on the performance analysis of o cooperative...

Blekinge Institute of TechnologyDoctoral Dissertation Series No. 2012:08

School of Computing

On the PerfOrmance analysis Of cOOPerative cOmmunicatiOns with Practical cOnstraints

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ISSN 1653-2090

ISBN 978-91-7295-235-5

With the rapid development of multimedia services,

wireless communication engineers may face a ma-

jor challenge to meet the demand of higher data-rate

communication over error-prone mobile radio chan-

nels. As a promising solution, the concept of coope-

rative communication, where a so-called relay node

is formed to assist the direct link, has recently been

applied to alleviate the severe pathloss and shado-

wing effects in wireless systems. In addition, wit-

hout spending extra spectrum and power resources,

multiple-input multiple-output (MIMO) antenna

systems have been shown to provide an immense

improvement in system performance compared to

its single-antenna counterpart. As such, cooperative

MIMO communication is essential for wireless and

mobile networks because of its remarkable increase

in spectral efficiency and reliability. Although the

utilization of cooperative communication in MIMO

systems has gained great attention in the literature,

most of the research works have assumed perfect

conditions. Inspired by the aforementioned discus-

sion, this thesis takes a step further to investigate the

performance of cooperative communications with

practical constraints. The thesis provides a general

framework for performance analysis of cooperative

communications subject to several practical cons-

traints such as antenna correlation, rank-deficiency

of the channel matrix, co-channel interference, and

interference-limited constraint of cognitive radio

networks based on an underlay spectrum-sharing

approach.

The thesis is divided into six parts. The first part

investigates the performance of orthogonal space-

time block codes (OSTBCs) over MIMO relay

networks in Nakagami-m fading channels under

the antenna correlation effect. The second part ex-

tends the full-rank MIMO channel to the case of the

MIMO channel matrix being of rank-deficiency. Se-

veral important findings on the impact of the sing-

le-keyhole effect (SKE) and double-keyhole effect

(DKE) are observed for two types of amplifying

mechanism at the relay, namely, linear and squaring

approaches. An important observation corroborated

by our studies is that for offering a tradeoff bet-

ween performance and complexity, we should use

the linear approach for SKE and the squaring ap-

proach for DKE. The third part generalizes the key-

hole effect to multi-keyhole channels. The exact and

asymptotic expressions for symbol error probability

(SEP) are derived for some specific cases such as

multi-keyhole MIMO/multiple-input single-output

(MISO) channel. The fourth part proposes a distri-

buted Alamouti space-time code for two-way fixed

gain amplify-and-forward (AF) relaying. In parti-

cular, closed-form expressions for approximated er-

godic sum-rate and exact pairwise error probability

(PWEP) are derived for Nakagami-m fading chan-

nels. To reveal further insights into array and diver-

sity gains, an asymptotic PWEP is also obtained.

The fifth part analyzes the outage performance of a

two-way fixed gain AF relay system with beamfor-

ming, arbitrary antenna correlation, and co-channel

interference (CCI). Finally, the sixth part investiga-

tes the impact of interference power constraint on

the performance of cognitive relay networks based

on the spectrum-sharing approach.

aBstract

2012:08

2012:08

Quang Trung Duong

On the Performance Analysis of Cooperative Communications with Practical Constraints

Quang Trung Duong

On the Performance Analysis of Cooperative Communications with Practical Constraints

Quang Trung Duong

Doctoral Dissertation in Telecommunications Systems

Blekinge Institute of Technology doctoral dissertation seriesNo 2012:08

School of ComputingBlekinge Institute of Technology

SWEDEN

2012 Quang Trung DuongSchool of ComputingPublisher: Blekinge Institute of Technology,SE-371 79 Karlskrona, SwedenPrinted by Printfabriken, Karlskrona, Sweden 2012ISBN: 978-91-7295-235-5ISSN 1653-2090urn:nbn:se:bth-00531

v

To my parents Ky and Thanh

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Abstract

With the rapid development of multimedia services, wireless communicationengineers may face a major challenge to meet the demand of higher data-ratecommunication over error-prone mobile radio channels. As a promising solu-tion, the concept of cooperative communication, where a so-called relay nodeis formed to assist the direct link, has recently been applied to alleviate the se-vere pathloss and shadowing effects in wireless systems. In addition, withoutspending extra spectrum and power resources, multiple-input multiple-output(MIMO) antenna systems have been shown to provide an immense improve-ment in system performance compared to its single-antenna counterpart. Assuch, cooperative MIMO communication is essential for wireless and mobilenetworks because of its remarkable increase in spectral efficiency and relia-bility. Although the utilization of cooperative communication in MIMO sys-tems has gained great attention in the literature, most of the research workshave assumed perfect conditions. Inspired by the aforementioned discussion,this thesis takes a step further to investigate the performance of coopera-tive communications with practical constraints. The thesis provides a generalframework for performance analysis of cooperative communications subjectto several practical constraints such as antenna correlation, rank-deficiencyof the channel matrix, co-channel interference, and interference-limited con-straint of cognitive radio networks based on an underlay spectrum-sharingapproach.

The thesis is divided into six parts. The first part investigates the per-formance of orthogonal space-time block codes (OSTBCs) over MIMO relaynetworks in Nakagami-m fading channels under the antenna correlation ef-fect. The second part extends the full-rank MIMO channel to the case of theMIMO channel matrix being of rank-deficiency. Several important findings onthe impact of the single-keyhole effect (SKE) and double-keyhole effect (DKE)are observed for two types of amplifying mechanism at the relay, namely, lin-ear and squaring approaches. An important observation corroborated by ourstudies is that for offering a tradeoff between performance and complexity,we should use the linear approach for SKE and the squaring approach forDKE. The third part generalizes the keyhole effect to multi-keyhole channels.The exact and asymptotic expressions for symbol error probability (SEP) arederived for some specific cases such as multi-keyhole MIMO/multiple-inputsingle-output (MISO) channel. The fourth part proposes a distributed Alam-outi space-time code for two-way fixed gain amplify-and-forward (AF) relay-ing. In particular, closed-form expressions for approximated ergodic sum-rate

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and exact pairwise error probability (PWEP) are derived for Nakagami-mfading channels. To reveal further insights into array and diversity gains, anasymptotic PWEP is also obtained. The fifth part analyzes the outage perfor-mance of a two-way fixed gain AF relay system with beamforming, arbitraryantenna correlation, and co-channel interference (CCI). Finally, the sixth partinvestigates the impact of interference power constraint on the performanceof cognitive relay networks based on the spectrum-sharing approach.

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Preface

This thesis summarizes my work within the field of cooperative communica-tions. The work has been performed at the School of Engineering and Schoolof Computing, Blekinge Institute of Technology, Karlskrona, Sweden. Thethesis consists of six parts:

Part I

Orthogonal Space-Time Block Codes with CSI-Assisted Amplify-and-Forward Relaying in Correlated Nakagami-m Fading Channels

Part II

Keyhole Effect in MIMO AF Relay Transmission with Space-Time BlockCodes

Part III

Multi-Keyhole Effect in MIMO AF Relay Downlink Transmission withSpace-Time Block Codes

Part IV

Distributed Space-Time Coding in Two-Way Fixed Gain Relay Net-works over Nakagami-m Fading Networks

Part V

Beamforming in Two-Way Fixed Gain Amplify-and-Forward Relay Sys-tems with CCI

Part VI Cognitive Cooperative Communication with Amplify-and-ForwardRelay and Spectrum-Sharing Approach

A Exact Outage Probability of Cognitive AF Relaying with UnderlaySpectrum Sharing

B Cooperative Spectrum Sharing Networks with AF Relay and Se-lection Diversity

C Effect of Primary Networks on the Performance of Spectrum Shar-ing AF Relaying

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Acknowledgements

It is now time to write some final sentences for my Ph.D. study, which hasbeen a joyful journey. When looking back to the last four and a half years,I am indebted to my principle advisor Prof. Hans-Jurgen Zepernick. Thisthesis would not have been completed without his guidance and support. Istill remember the first time we met at IEEE VTC-Fall conference 2007 inBaltimore, USA. Afterwards, he offered me a Ph.D. fellowship to pursue mypostgraduate study at Blekinge Institute of Technology (BTH), Sweden. I amlucky to have him as my supervisor. To me, he is the best advisor that a Ph.D.student can ask for. I have learnt many useful things from him: patience andpassion. His professional skills have lifted me to a level that I am now today.Working with him is one of few privileges in my life.

During my Ph.D. study, I was lucky to meet many experts in the field.One of them is my co-advisor, Prof. Markus Fiedler at BTH. I admire him forhis deep knowledge and expertise in Quality-of-Experience, cross-layer designfor mobile multimedia applications. From his guidance, I figured out that Ican collaborate with other researchers whose interests are different from mine.

This thesis topic is cooperative communications and it is even more mean-ingful to have great cooperation from other scholars and friends: Prof. Aru-mugam Nallanathan (King’s College London, UK), Dr. Himal A. Suraweera(Singapore University of Technology and Design, Singapore), Prof. TheodorosA. Tsiftsis (Technological Educational Institute of Lamia, Greece), Dr. VoNguyen Quoc Bao (Posts and Telecommunications Institute of Technology,Vietnam), Prof. Kyeong Jin Kim (Inha University, Korea). A devout thankgoes to Prof. Nallanathan for his support and guidance. The warmest thankto Himal for so many interesting daily talks, I have learnt a lot from his ma-turity and expertise. Many thanks to Theo, the first researcher that I havecollaborated with, for showing me how to make international collaborationsuccessful. Special thanks to Bao, I still recalled the first joint work, our ELpaper, is a huge driving-force for my research, which gave me a lot of confi-dence to continue the challenging journey of my academic life. Many thanksKeyongjin for helpful discussions about cyclic-prefix single-carrier systems. Iam thankful to all of you for both technical and non-technical issues, whichbring to my life wonderful friends and colleagues.

Throughout these years, it has been an excellent opportunity to visit otherinstitutions. Deepest thanks go to Prof. Yao Wang, Prof. Elza Erkip, andProf. Chau Yuen for giving me an opportunity to join their research groupas a visiting scholar at Polytechnic Institute of New York University in 2009

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and Singapore University of Technology and Design in 2011. I would alsolike to thank Prof. Yan Zhang (Simula Research Laboratory, Norway), Dr.Yuexing Peng (Beijing University of Posts and Telecommunications, China),Dr. Lei Shu (Osaka University, Japan), Prof. Magnus Jonsson (HalmstadUniversity, Sweden) for inviting me to present my works. Special thanks goto the Knowledge Foundation (KK-Stiftelsen) for funding this research.

I am also thankful to all of my co-authors for many fruitful discussion andcollaboration over these years. My gratefulness goes to Dr. Maged Elkashlan(Queen Mary University of London, UK), Dr. George C. Alexandropoulos(Athens Information Technology, Greece), Dr. Phee Lep Yeoh (Universityof Melbourne, Australia), Dr. Nan Yang (University of New South Wales,Australia), Dr. Daniel Benevides da Costa (Federal University of Ceara,Brazil), Dr. Fawaz S. Al-Qahtani (Texas A&M University at Qatar, Qatar),Dr. Nguyen-Son Vo (Huazhong University of Science and Technology, China),Mr. Hien Quoc Ngo (Linkoping University, Sweden), Dr. Xianfu Lei (South-west Jiaotong University, China). I would like to thank my other colleaguesand friends at the Radio Communications Group, the School of Engineering,and the School of Computing at BTH for making my stay in Karlskrona andRonneby more enjoyable.

Finally, I would like to thank my parents Ky and Thanh, my brother Huy,my aunt Melodie, my uncle Tom for always being there for me, for givingme their permanent love and support, which makes me the luckiest son andnephew. Last but certainly not least, I want to thank my wife Hien for herlove and encouragement. Thank you for always believing in me.

Quang Trung DuongKarlskrona, May 2012

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Publication List

Part I is published as

T. Q. Duong, G. C. Alexandropoulos, T. A. Tsiftsis, and H.-J. Zepernick, “Or-thogonal Space-Time Block Codes with CSI-Assisted Amplify-and-ForwardRelaying in Correlated Nakagami-m Fading Channels,” IEEE Trans. on Veh.

Techno., vol. 60, no. 3, pp. 882–889, Mar. 2011.

Based on

T. Q. Duong, H.-J. Zepernick, T. A. Tsiftsis, and V. N. Q. Bao, “Amplify-and-Forward MIMO Relaying with OSTBC over Nakagami-m Fading Channels,”in Proc. IEEE International Communications Conference, Cape Town, SouthAfrica, pp. 1–6, May 2010.

T. Q. Duong, G. C. Alexandropoulos, T. A. Tsiftsis, and H.-J. Zepernick,“Outage Probability of MIMO AF Relay Networks over Nakagami-m FadingChannels,” Electron. Lett., vol. 46, no. 17, pp. 1229–1231, Sep. 2010.

Part II is published as

T. Q. Duong, H. A. Suraweera, T. A. Tsiftsis, H.-J. Zepernick, and A. Nal-lanathan, “Keyhole Effect in MIMO AF Relay Transmission with Space-TimeBlock Codes,” IEEE Trans. Commun., Feb. 2012, under revision.

Based on

T. Q. Duong, H. A. Suraweera, T. A. Tsiftsis, H.-J. Zepernick, and A. Nal-lanathan, “OSTBCs in MIMO AF Relay Systems with Keyhole and Corre-lation Effects,” in Proc. IEEE International Communications Conference,Kyoto, Japan, pp. 1–6, Jun. 2011.

Part III is published as

T. Q. Duong, H. A. Suraweera, C. Yuen, and H.-J. Zepernick, “Multi-KeyholeEffect in MIMO AF Relay Downlink Transmission with Space-Time BlockCodes,” in Proc. IEEE Global Communications Conference, Houston, TX,pp. 1–6, Dec. 2011.

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Part IV is published as

T. Q. Duong, H. Q. Ngo, H.-J. Zepernick, A. Nallanathan, “Distributed Space-Time Coding in Two-Way Fixed Gain Relay Networks over Nakagami-m Fad-ing Networks,” in Proc. IEEE International Communications Conference,Ottawa, Canada, June 2012.

Based on

T. Q. Duong, C. Yuen, H.-J. Zepernick, X. Lei, “Average Sum-Rate of Dis-tributed Alamouti Space-Time Scheme in Two-Way Amplify-and-Forward Re-lay Networks,” in Proc. IEEE Global Communications Conference Workshop,Miami, FL, pp. 79–83, Dec. 2010.

Part V is published as

T. Q. Duong, H. A. Suraweera, H.-J. Zepernick, C. Yuen, “Beamformingin Two-Way Fixed Gain Amplify-and-Forward Relay Systems with CCI,”in Proc. IEEE International Communications Conference, Ottawa, Canada,June 2012.

Part VI is published as

T. Q. Duong, V. N. Q. Bao, H. Tran, G. C. Alexandropoulos, and H.-J. Zeper-nick, “Effect of Primary Networks on the Performance of Spectrum SharingAF Relaying,” Electron. Lett., vol. 48, no. 1, pp. 25–27, Jan. 2012.

T. Q. Duong, V. N. Q. Bao, G. C. Alexandropoulos, and H.-J. Zepernick,“Cooperative Spectrum Sharing Networks with AF Relay and Selection Di-versity,” Electron. Lett., vol. 47, no. 20, pp. 1149–1151, Sep. 2011.

T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick, “Exact Outage Probabilityof Cognitive AF Relaying with Underlay Spectrum Sharing,” Electron. Lett.,vol. 47, no. 47, pp. 1001-1002, Aug. 2011.

Publications in conjunction with this thesis but not included:

Book and Book Chapters

Quang Trung Duong, “On Cooperative Communications and Its Applicationsto Mobile Multimedia,” Blekinge Institute of Technology, Karlskrona, Sweden,

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Licentiate Thesis, Apr. 2010.

T. Q. Duong and H.-J. Zepernick, “Cross-Layer Design for Packet Data Trans-mission in Co-located MIMO Systems,” Chapter 16 in the book “Using Cross-

Layer Techniques for Communication Systems Techniques and Applications”IGI Publisher, 2012.

T. Q. Duong, N.-S. Vo, H.-J. Zepernick, et al., “Replication Strategies forVideo On-Demand over Wireless Mesh Networks: A Cross-Layer Optimiza-tion Approach,” Chapter 15 in the book “Using Cross-Layer Techniques for

Communication Systems Techniques and Applications” IGI Publisher, 2012.

Journals

T. Q. Duong and H.-J. Zepernick, “Cross-Layer Design for MRT Systems withChannel Estimation Error and Feedback Delay,” Wireless Personal Commu-

nications, vol. 58, no. 4, pp. 681–694, 2010.

T. Q. Duong, H.-J. Zepernick, and V. N. Q. Bao, “Symbol Error Probabilityof Hop-by-Hop Beamforming in Nakagami-m Fading,” Electron. Lett., vol. 44,no. 20, pp. 1206–1207, Sep. 2009.

T. Q. Duong and H.-J. Zepernick, “On the Performance Gain of HybridDecode-Amplify-Forward Cooperative Communications,” EURASIP Journal

on Wireless Communications and Networking, vol. 2009, article ID 479463,10 pages, 2009. doi:10.1155/2009/479463.

T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick, “On the Performance of Se-lection Decode-and-Forward Relay Networks over Nakagami-m Fading Chan-nels,” IEEE Commun. Lett., vol. 13, no. 3, pp. 172–174, Mar. 2009.

Conferences

T. Q. Duong, O. Alay, E. Erkip, and H.-J. Zepernick, “End-to-End Perfor-mance of Randomized Distributed Space-Time Codes,” in Proc. IEEE Per-

sonal, Indoor and Mobile Radio Communications, Istanbul, Turkey, pp. 988–993, Sep. 2010.

T. Q. Duong, H.-J. Zepernick, T. A. Tsiftsis, and V. N. Q. Bao, “PerformanceAnalysis of Amplify-and-Forward MIMO Relay Networks with Transmit An-tenna Selection over Nakagami-m Channels,” in Proc. IEEE Personal, Indoor

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and Mobile Radio Communications, Istanbul, Turkey, pp. 368–372, Sep. 2010.

T. Q. Duong, U. Engelke, and H.-J. Zepernick, “Cooperative Wireless Com-munications with Unequal Error Protection and Fixed Decode-and-ForwardRelays,” in Proc. International Conference on Communications and Electron-

ics, Nha Trang, Vietnam, pp. 702–706, Aug. 2010.

T. Q. Duong, H.-J. Zepernick, and M. Fiedler, “Cross-Layer Design for In-tegrated Mobile Multimedia Network with Strict Priority Traffic,” in Proc.

IEEE Wireless Communications and Networking Conference, Sydney, Aus-tralia, pp. 1–6, Apr. 2010.

T. Q. Duong and H.-J. Zepernick, “Performance Analysis of Cooperative Spa-tial Multiplexing with Amplify-and-Forward Relays,” in Proc. IEEE Per-

sonal, Indoor and Mobile Radio Communications, Tokyo, Japan, pp. 1963–1967, Sep. 2009.

T. Q. Duong and H.-J. Zepernick, “Adaptive Transmission Scheme for Wire-less Cooperative Communications,” in Proc. IEEE Personal, Indoor and Mo-

bile Radio Communications, Tokyo, Japan, pp. 1958–1962, Sep. 2009.

T. Q. Duong and H.-J. Zepernick, ‘Hybrid Decode-Amplify-Forward Cooper-ative Communications with Multiple Relays,” in Proc. IEEE Wireless Com-

munications and Networking Conference, Budapest, Hungary, pp. 1–6, Apr.2009.

T. Q. Duong and H.-J. Zepernick, “On the Ergodic Capacity of CooperativeSpatial Multiplexing Systems in Composite Channels,” in Proc. IEEE Radio

and Wireless Symposium, San Diego, CA, pp. 175–178, Jan. 2009.

T. Q. Duong, U. Engelke, and H.-J. Zepernick, “Unequal Error Protection forWireless Multimedia Transmission in Decode-and-Forward Relay Networks,”in Proc. IEEE Radio and Wireless Symposium, San Diego, CA, pp. 703–706,Jan. 2009, (finalists for the Student’s Best Paper Competition).

T. Q. Duong and H.-J. Zepernick, “Average Symbol Error Rate of Cooper-ative Spatial Multiplexing in Composite Channels,” in Proc. IEEE Interna-

tional Symposium on Wireless Communication Systems, Reykjavik, Iceland,pp. 335–339, Oct. 2008.

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T. Q. Duong and H.-J. Zepernick, “Robust EZW Image Transmission SchemeUsing Distributed-Alamouti Codes in Relay Networks,” in Proc. International

Conference on Signal Processing and Communication Systems, Gold Coast,Australia, pp. 1–6, Dec. 2008.

T. Q. Duong and H.-J. Zepernick, “On the Performance of ROI-Based Im-age Transmission Using Cooperative Diversity,” in Proc. IEEE International

Symposium on Wireless Communication Systems, Reykjavik, Iceland, pp.340–343, Oct. 2008.

Other publications:

Journals

K. J. Kim, T. Q. Duong, and H. V. Poor, “Performance Analysis of AdaptiveDecode-and-Forward Cooperative Single-Carrier Systems,” IEEE Trans. on

Veh. Technol., Apr. 2012 (accepted).

T. Q. Duong, D. B. da Costa, M. Elkashlan, and V. N. Q. Bao, “Cogni-tive Amplify-and-Forward Relay Networks over Nakagami-m Fading,” IEEE

Trans. on Veh. Technol., 2012 (in press).

H. Phan, T. Q. Duong, and H.-J. Zepernick, “Performance Analysis of Decoup-le-and-Forward MIMO Relaying in Nakagami-m Fading,” IEICE Trans on

Communications, May 2012, accepted.

H. Phan, T. Q. Duong, H.-J. Zepernick, and L. Shu, “Adaptive Transmissionin MIMO AF Relay Networks with Orthogonal Space-Time Block Codes overNakagami-m Fading,” EURASIP Journal on Wireless Communications and

Networking, vol. 2012:11, 2012. doi:10.1186/1687-1499-2012-11.

H. Tran, T. Q. Duong, and H.-J. Zepernick, “Delay Performance of Cogni-tive Radio Networks for Point-to-Point and Point-to-Multipoint Communica-tions,” EURASIP Journal on Wireless Communications and Networking, vol.2012:9, 15 pages, 2012. doi:10.1186/1687-1499-2012-9.

H. Phan, T. Q. Duong, M. Elkashlan, and H.-J. Zepernick, “BeamformingAmplify-and-Forward Relay Networks with Feedback Delay and Interference,”IEEE Sig. Process. Lett., vol. 19, no. 1, pp. 16–19, Jan. 2012.

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M. Yan, Q. Chen, X. Lei, T. Q. Duong, and P. Fan, “Outage Probabilityof Switch and Stay Combining in Two-way Amplify-and-Forward Relay Net-works,” IEEE Wireless Commun. Lett., Feb. 2012, (in press).

V. N. Q. Bao and T. Q. Duong, “Exact Outage Probability of CognitiveUnderlay DF Relay Networks with Best Relay Selection,” IEICE Trans on

Communications, vol. E95-B, no. 6, Jun. 2012.

M. Elkashlan, P. L. Yeoh, N. Yang, T. Q. Duong, and C. Leung, “A Com-parison of Two MIMO Relaying Protocols in Nakagami-m Fading Channels,”IEEE Trans. on Veh. Technol., vol. 61, no. 3, Mar. 2012.

V. N. Q. Bao and T. Q. Duong, “Outage Analysis of Cognitive Multihop Net-works under Interference Constraints,” IEICE Trans. on Communications,vol. E95-B, no. 3, pp. 1019–1022, Mar. 2012.

F. S. Al-Qahtani, T. Q. Duong, C. Zhong, A. Qaraqe, and H. Alnuweiri,“Performance Analysis of AF Dual-Hop Relaying Systems over Nakagami-mFading Channels in the Presence of Interference at the Relay,” IEEE Sig.

Process. Lett., vol. 60, no. 3, pp. 882–889, Mar. 2011.

J. Yang, P. Fan, T. Q. Duong, and X. Lei, “Exact Performance of Two-WayAF Relaying in Nakagami-m Fading Environment,” IEEE Trans. on Wireless

Commun., vol. 10, no. 3, pp. 980–987, Mar. 2011.

T. Q. Duong, L.-N. Hoang, and V. N. Q. Bao, “On the Performance of Two-Way Amplify-and-Forward Relay Networks,” IEICE Transactions on Com-

munications, vol. Vol.E92-B, no. 12, pp. 3957–3959, Dec. 2009.

T. Q. Duong, N.-T. Nguyen, T. Hoang, and V.-K. Nguyen, “Pairwise ErrorProbability of Distributed Space-Time Coding Employing Alamouti Schemein Wireless Relays Networks,” Springer Wireless Personal Communications,vol. 51, no. 2, pp. 231–244, Oct. 2009.

T. Q. Duong and V. N. Q. Bao, “Performance Analysis of Selection Decode-and-Forward Relay Networks,” Electron. Lett., vol. 44, no. 20, pp. 1206–1207,Sep. 2008.

T. Q. Duong, H. Shin, and E.-K. Hong, “Error Probability of Binary and M-ary Signals with Spatial Diversity in Nakagami-q (Hoyt) Fading Channels,”EURASIP Journal on Wireless Communications and Networking, vol. 2007,

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article ID 53742, 8 pages, 2007. doi:10.1155/2007/53742.

Conferences

C.-T.-M. Chinh, T. Q. Duong, and H.-J. Zepernick, “MRT/MRC for Cog-nitive AF Relay Networks Under Feedback Delay and Channel EstimationError,” in Proc. IEEE Personal, Indoor and Mobile Radio Communications,Sydney, Australia, Sep. 2012.

N.-S. Vo, T. Q. Duong, and L. Shu, “Bit Allocation for Multi-Source Multi-Path P2P Video Streaming in VoD Systems over Wireless Mesh Networks,”in Proc. IEEE International Communications Conference, Ottawa, Canada,June 2012.

M. V. Nguyen, C. S. Hong, and T. Q. Duong, “Joint Optimal Rate, Power, andSpectrum Allocation in Multi-hop Cognitive Radio Networks,” in Proc. IEEE

International Communications Conference, Ottawa, Canada, June 2012.

H. Phan, T. Q. Duong, and H.-J. Zepernick, “MIMO Cooperative Multiple-Relay Networks with OSTBCs over Nakagami-m Fading Channels,” in Proc.

IEEE Wireless Communications and Networking Conference, Paris, France,Apr. 2012.

H. Tran, T. Q. Duong, and H.-J. Zepernick, “Performance Analysis of Cogni-tive Relay Networks Under Power Constraint of Multiple Primary Users,” inProc. IEEE Global Communications Conference, Houston, TX, pp. 1–6, Dec.2011.

H. Phan, T. Q. Duong, and H.-J. Zepernick, “MIMO AF Semi-Blind Re-lay Networks with OSTBC Transmission over Nakagami-m Fading,” in Proc.

IEEE International Conference Signal Processing and Communication Sys-

tems, Honolulu, HI, pp. 1–5, Dec. 2011

H. Phan, T. Q. Duong, and H.-J. Zepernick, “Outage Performance for Op-portunistic Decode-and-Forward Relaying Coded Cooperation Networks overNakagami-m Fading,” in Proc. IEEE International Symposium on Wireless

Communications Systems, Aachen, Germany, pp. 417–421, Nov. 2011.

H. Tran, T. Q. Duong, and H.-J. Zepernick, “On the Performance of Spec-trum Sharing Systems over Alpha-Mu Fading Channels for Non-identical MuParameters,” in Proc. IEEE International Symposium on Wireless Commu-

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nications Systems, Aachen, Germany, pp. 477–481, Nov. 2011.

H. Q. Ngo, T. Q. Duong, and E. G. Larsson, “Uplink Performance Analysis ofMulticell MU-MIMO with Zero-Forcing Receivers and Perfect CSI,” in Proc.

IEEE Swedish Communications Technology Workshop, Stockholm, Sweden,pp. 40–45, Oct. 2011.

C.-T.-M. Chinh, T. Q. Duong, and H.-J. Zepernick, “Outage Probability andErgodic Capacity for MIMO-MRT Systems under Co-Channel Interferenceand Imperfect CSI,” in Proc. IEEE Swedish Communications Technology

Workshop, Stockholm, Sweden, pp. 46–51, Oct. 2011.

H. Phan, T. Q. Duong, and H.-J. Zepernick, “SER of Amplify-and-ForwardCooperative Networks with OSTBC Transmission in Nakagami-m Fading,” inProc. IEEE Vehicular Technology Conference Fall, San Francisco, CA, pp. 1–5, Sep. 2011.

S.-N. Vo, T. Q. Duong, H.-J. Zepernick, L. Shu and X. Du, “Cross-LayerDesign for Video Replication Strategy over Multihop Wireless Networks,” inProc. IEEE International Communications Conference, Kyoto, Japan, pp. 1–6, Jun. 2011.

T. Q. Duong, T.-T. Le, and H.-J. Zepernick, “Performance of Cognitive RadioNetworks with Maximal Ratio Combining over Correlated Rayleigh Fading,”in Proc. International Conference on Communications and Electronics, NhaTrang, Vietnam, pp. 65–69, Aug. 2010.

H. Tran, T. Q. Duong, and H.-J. Zepernick, “Average Waiting Time of Pack-ets with Different Priorities in Cognitive Radio Networks,” in Proc. IEEE

International Symposium on Wireless Pervasive Computing, Modena, Italy,pp. 122–127, May 2010.

H. Phan, T. Q. Duong, and H.-J. Zepernick, “Full-Rate Distributed Space–Time Coding for Bi-Directional Cooperative Communications,” in Proc. IEEE

International Symposium on Wireless Pervasive Computing, Modena, Italy,pp. 22–26, May 2010.

N.-N. Tran and T. Q. Duong, “Training Design Upon Mutual Information forSpatially Correlated MIMO-OFDM,” in Proc. IEEE Wireless Communica-

tions and Networking Conference, Sydney, Australia, Apr. 2010.

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F. Al-Qahatani and T. Q. Duong, “Selection Decode-and-Forward Relay Net-works with Rectangular QAM in Nakagami-m Fading Channels,” in Proc.

IEEE Wireless Communications and Networking Conference, Sydney, Aus-tralia, pp. 1–4, Apr. 2010.

V. N. Q. Bao, T. Q. Duong, and N.-N. Tran, “Ergodic Capacity of Cooper-ative Networks Using Adaptive Transmission and Selection Combining,” inProc. International Conference Signal Processing and Communication Sys-

tems, Nebraska, NE, pp. 1–6, Oct. 2009.

T. Q. Duong, “Exact Closed-Form Expression for Average Symbol Error Rateof MIMO-MRC Systems,” in Proc. International Conference on Advanced

Technologies for Communication, Hanoi, Vietnam, pp. 20–23, Oct. 2008.

T. Q. Duong, D.-B. Ha, H.-A. Tran, and N.-S. Vo, “Symbol Error Probabilityof Distributed-Alamouti Scheme in Wireless Relay Networks,” in Proc. IEEE

67th Vehicular Technology Conference Spring, Singapore, pp. 648–652, May2008.

T. Q. Duong and H.-A. Tran, “Distributed Space-Time Block Codes with Am-icable Orthogonal Designs,” in Proc. IEEE Radio and Wireless Symposium,Orlando, FL, pp. 559–562, Jan. 2008.

xxiii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Publications list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part I

Orthogonal Space-Time Block Codes with CSI-Assisted Amplify-and-Forward Relaying in Correlated Nakagami-m Fading Channels . . . . 33

Part II

Keyhole Effect in MIMO AF Relay Transmission with Space-Time BlockCodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Part III

Multi-Keyhole Effect in MIMO AF Relay Downlink Transmission withSpace-Time Block Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Part IV

Distributed Space-Time Coding in Two-Way Fixed Gain Relay Net-works over Nakagami-m Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Part V

Beamforming in Two-Way Fixed Gain Amplify-and-Forward Relay Sys-tems with CCI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Part VI

Cognitive Cooperative Communication with Amplify-and-Forward Re-lay and Spectrum-Sharing Approach

A Exact Outage Probability of Cognitive AF Relaying with UnderlaySpectrum Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xxiv

B Cooperative Spectrum Sharing Networks with AF Relay and Se-lection Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

C Effect of Primary Networks on the Performance of Spectrum Shar-ing AF Relaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

xxv

Abbreviations

c.n.i.d. correlated but not necessarily identically distributedi.n.i.d. independent but not necessarily identically distributedi.i.d. independent and identically distributedAF Amplify-and-ForwardAWGN Additive White Gaussian NoiseBER Bit Error RateBPSK Binary Phase-Shift KeyingBRS Best Relay SelectionCCI Co-Channel InterferenceCDF Cumulative Distribution FunctionCRN Cognitive Radio NetworksCSI Channel State InformationDASTC Distributed-Alamouti Space-Time CodeDF Decode-and-ForwardDKE Double Keyhole EffectDSTC Distributed Space-Time CodeMGF Moment Generating FunctionMIMO Multiple-Input Multiple-OutputMISO Multiple-Input Single-OutputML Maximum-LikelihoodMRC Maximal-Ratio CombiningOP Outage ProbabilityOSTBC Orthogonal Space-Time Block CodePDF Probability Density FunctionPRS Partial Relay SelectionPU Primary UserPWEP Pairwise Error ProbabilitySEP Symbol Error ProbabilitySINR Signal-to-Interference Plus Noise RatioSISO Single-Input Single-OutputSNR Signal-to-Noise RatioSKE Single Keyhole EffectSU Secondary UserTAS Transmit Antenna Selection

Introduction

1 Motivation

With the ever increasing demand of multimedia services, future wireless gen-erations aim to achieve higher data rates and more reliable communicationsfor Quality of Service (QoS) provision. However, due to multipath fading,severe shadowing, pathloss, and co-channel interference (CCI), communica-tion in single-hop wireless networks has faced some fundamental limits [1].In order to alleviate the impairment inflicted by wireless channels, multiple-input multiple-output (MIMO) systems have been proposed to exploit therich-scattering nature of multiple-antenna channels [2–7]. By deploying mul-tiple antennas at the transceivers, the diversity and multiplexing gains haveincreased up to nSnD and min(nS, nD), where nS and nD are the number ofantennas at the source and destination, respectively. In comparison to itssingle-antenna counterpart, this enhancement of MIMO systems is remark-able as single-input single-output (SISO) systems are known to only achieveunit order of diversity and multiplexing gains. As a result, the MIMO tech-nology has been adopted to many commercial standards [8]. The concept ofMIMO systems in single-hop has been intensively studied in many aspects,e.g., capacity [9,10], spatial diversity [11], space-time code design [12,13], andinformation theory [14]. However, due to the error-prone wireless channels,MIMO communication over a single-hop transmission may not be feasible.As a result, an alternative communication technique for wireless networks,namely cooperative communication, is required, where the signals from thesource should traverse through multiple intermediate terminals.

The basic concept of cooperative communication/relay networks is to uti-lize the assistance of relay nodes for conveying the source’s messages to thedestination. In particular, the transmission between the source and desti-nation nodes is divided into two main phases: 1) Broadcasting phase: thesource transmits its messages to both relay and destination, and 2) Multiple-

1

2 Introduction

access phase: the relay manipulates its received messages from the sourcebefore forwarding them to the destination. With this transmission strategy,cooperative communications can overcome the severe pathloss and shadowingeffects. As a result, the concept of cooperative communications has gainedgreat attention, inspired by the pioneering works [15–17]. It has been shownthat cooperative communications can achieve significant power savings forextending network life-time, expand the communication range, and keep theimplementation complexity low [18–21].

Although extensive research has been made on cooperative communica-tions, most of the previous works have assumed perfect conditions. As a re-sult, the contribution of this thesis is to aim at investigating the performanceof cooperative communications in realistic environments, e.g., the impact ofantenna correlation at the transmit and received sides, the lack of scatteringobjects in MIMO relay channels (keyhole/multi-keyhole effects), the existenceof co-channel interference from adjacent cells.

In addition, the inefficiency of spectrum utilization can be alleviated byusing the cognitive radio concept, where secondary networks can co-occupyfrequency bands which are licensed to a primary network. However, the cog-nitive network performance is decreased as the transmit power is limited sothat the secondary signal does not cause any harmful interference on primarynetworks. As a result, utilizing relaying can significantly improve the cogni-tive network performance. Our main target in this thesis is to investigate theimpact of peak interference power constraint imposed by the primary receiversand the interfering power from primary transmitters on the performance ofcognitive relay networks.

2 An Overview on Cooperative Communica-

tions

2.1 Background

The concept of cooperative communications was first introduced in [22], wherethe capacity for the three-terminal communication shown in Fig. 1 was stud-ied. Since 2000, this concept has gained great attention in the research com-munity [15–21, 23–25]. Depending on the relaying operation, the relay canbe mainly categorized into two schemes: i) decode-and-forward (DF) and ii)amplify-and-forward (AF), each of which has its own advantages and disad-vantages. For the DF scheme, the relay is required to perform an extra opera-

Introduction 3

tion by decoding the source signal before forwarding it to the destination [19].In contrast, for the AF scheme, the relay simply amplifies the received mes-sage with a scalar gain without performing any signal regeneration, whichmay cause noise accumulation at the destination.

Relay

Source Destination

S

R

D

Figure 1: Basic relay network: A source S communicates with a destinationD through the assistance of a relay R.

In AF relaying, depending on how the scalar gain is generated, the re-lay can be further classified as variable-gain, in which the full knowledge onchannel state information (CSI) of the first hop is required, and fixed-gain,i.e., only the statistical distribution of the fading of the first hop is needed.The performance of variable-gain AF relaying over Rayleigh and Nakagami-m fading channels has been reported in [26, 27]. The work in [27] is limitedto the harmonic mean of two gamma random variables (RVs), which causesthe final expressions not to be an exact closed-form but lower bound. Theexact closed-form expressions of the outage probability (OP) and bit errorrate (BER) for fixed-gain AF relaying have been obtained for Rayleigh fadingchannels in [28]. Tsiftsis et al. have derived exact closed-form expressions ofboth variable-gain and fixed-gain AF relaying over Nakagami-m fading chan-nels in [29]. Moreover, the impact of the direct link has been considered byapplying selection combining at the destination. It is important to note thatthe works of Hasna and Alouini in [26,27,27] and of Tsiftsis et al. in [29] haveinspired a series of research papers which will be addressed in the sequel.

4 Introduction

2.2 Cooperative Communications with Single-Antenna

Terminals

2.2.1 Multi-hop Transmission

The use of relaying, previously mainly focusing on dual-hop, has been ex-tended to multi-hop communication by allowing the signal to traverse throughthe multiple intermediate nodes, as shown in Fig. 2. It has been shownin [26,27,30–33] that multi-hop relay networks significantly improve the com-munication coverage of cellular and ad hoc networks without spending extranetwork resources such as bandwith and power. The performance analysis forfixed-gain AF multi-hop relaying has been first proposed for Rayleigh fadingin [34]. In [35], the authors have studied the performance bounds for multi-hop transmissions with fixed-gain AF relaying over several important fadingmodels such as Nakagami-n (Rice), Nakagami-q (Hoyt), and Nakagami-m fad-ing channels. For Nakagami-m fading channels, the OP of variable-gain AFmulti-hop relaying has been addressed in [36].

Source Destination

Hop 1 Hop 2 Hop N

R1S R2 Rk RN-1 D

Figure 2: Cooperative communications system with multiple relays: Multi-hop transmission.

2.2.2 Dual-hop Transmission with Multiple Relays

It is important to note that one of the main challenges for multiple-relaycommunications is the synchronization between different distributed termi-nals, which requires a centralized controller. The best relay selection (BRS),proposed by Bletsas et al. in [37], has been considered as the simplest relayingcombining strategy for achieving the full diversity, as shown in Fig. 3. As such,the performance of the BRS scheme has been extensively studied [38–42]. Ithas been shown in [43] that the opportunistic relaying with DF relays achievesthe global optimum in OP performance. The performance of BRS transmis-sion has been derived for DF relays in [38, 39, 42]. The work in [43] has beenextended to opportunistic AF relaying, where an exact closed-form expressionfor OP has been obtained [44]. For Nakagami-m fading channels, Duong et

Introduction 5

al. have derived closed-form expressions for outage probability, symbol errorprobability (SEP), and ergodic capacity in [41]. Recently, the performanceof multiple relays where the destination deploys the maximal-ratio combin-ing (MRC) technique has been derived in [45]. The aforementioned works onBRS require the full CSI knowledge of the dual-hop. A simpler version ofBRS, namely partial relay selection (PRS), has been considered to reduce thecomplexity and signaling overhead for relay communication [46]. AlthoughPRS reduces the performance compared with BRS, by assuming that onlythe CSI of the first hop is needed, its simple implementation has attractedincreasing interest [47–53]. The performance of PRS fixed-gain AF relays forRayleigh and Nakagami-m fading channels have been reported in [47] and [52],respectively. The asymptotic result for PRS variable-gain AF relays has beenobtained in [48]. An important impact on the performance of PRS, namelyfeedback delay, has been investigated in [51].

Source Destination

R1

S

R2

Rk

RN

D

N Relays

Figure 3: Cooperative communications system with multiple relays: Bestrelay selection (solid line: selected relay).

6 Introduction

2.2.3 Two-Way Relaying

Transmission of cooperative communications has to take place over dual-hopsor multi-hops, which significantly reduces the spectral efficiency of relay net-works compared to its single-hop counterpart. This loss can be recovered byutilizing two-way relay networks (e.g., see [54–56]), where the communicationbetween multiple sources occurs in two-way or multi-way transmission, asshown in Fig. 4. Particularly, two source nodes simultaneously transmit theirinformation to the relay node in the multiple-access phase (solid lines). Then,the relay node broadcasts the received signal to the two source nodes in thebroadcast phase (dotted lines) [57]. Exact closed-form expressions for outageprobability, SEP, and ergodic capacity for two-way AF relay networks overRayleigh fading channels have been introduced in [58]. This work has beenextended to Nakagami-m fading channels with the fading severity parameterm being integer or integer plus 0.5 [59]. The relay selection for two-way AFrelaying has been considered in [60].

S1 R S2

Figure 4: Cooperative communications system with multiple sources: Two-way transmission (multiple-access phase: solid lines; broadcast phase: dottedlines).

2.2.4 Coded Cooperation and Distributed Space-Time Coding

Coded cooperation, a combination between channel coding and relay coop-eration, is also promising to improve system performance [61–63]. Besideschannel coding, space-time codes can be also utilized for cooperative com-munications. Here, each relay can act as a virtual antenna to form thespace-time code in a distributed fashion, namely distributed space-time code(DSTC) [18, 64–68]. Specifically, a group of single-symbolwise maximum-likelihood (ML) decoding DSTCs with full diversity order has been proposed

Introduction 7

in [69]. However, these DSTCs contain a large number of zero entries in theDSTC design, which may yield a large peak power for non-linear amplifier atthe transmitter. In [70], the existing orthogonal and quasi-orthogonal designsfor co-located MIMO systems have been applied in AF relay networks. It hasbeen demonstrated in [70] that these DSTCs can achieve full diversity orderwith single-symbolwise ML decoding complexity. However, double number oftime-slots for the first-hop transmission is needed which leads to a decreaseof data rate compared to the proposed scheme in [69].

3 MIMO Relay Networks

MIMO systems offer efficient solutions to increase the reliability and datarate of wireless networks by deploying multiple antennas at both ends. Re-lay/cooperative communication has been considered to provide the benefitof extending coverage of wireless networks when the direct link may not beapplicable. As such, combining MIMO and relay concepts has drawn greatattention in recent years (see, e.g., [25,71] and the references therein). Specif-ically, the potential application of MIMO relay systems shown in Fig. 5 hasbeen presented and several bounds of the ergodic capacity over Rayleigh fad-ing channels have been provided in [25]. Several important MIMO AF re-lay schemes have been applied, for example: 1) transmit antenna selection(TAS) at the transmitter and MRC at the receiver, i.e., TAS/MRC sys-tems [72–79], 2) end-to-end best antenna selection in two-hop systems, i.e.,TAS/TAS systems [80–85], and 3) hop-by-hop beamforming, i.e., MRC/MRCsystems [86–90].

S R D

nS nR nR nD

Figure 5: Basic MIMO cooperative communication system.

8 Introduction

The performance of TAS/MRC over Rayleigh fading channels has beenreported in [72–74]. The exact and asymptotic SEP of TAS/MRC over inde-pendent and identically distributed (i.i.d.) Nakagami-m fading with multipleantennas being deployed at all terminals have been derived for fixed-gain andvariable-gain AF relaying in [77] and [78], respectively. The OP and SEPof TAS/MRC over independent but not necessarily identically distributed(i.n.i.d.) Nakagami-m fading with a single-antenna relay have been reportedin [79].

By assuming that perfect CSI is available at both the transmitter andreceiver, the TAS scheme has been utilized for both hops in MIMO relaysystems. The performance of TAS/TAS over Rayleigh fading channels indual-hop relay networks has been shown in [80–82]. Then, these works havebeen extended to the case of multi-hop Rayleigh fading channels in [83].Later, closed-form expressions for the exact and asymptotic OP and SEPof TAS/TAS over the Nakagami-m have been derived in [84, 85]. The com-parison between TAS/MRC and TAS/TAS has been reported in [91]. It hasbeen shown that the two schemes exhibit the same diversity order.

Another scheme to achieve the full diversity gain while keeping the com-plexity low is hop-by-hop beamforming, i.e., MRC/MRC systems. The per-formance of hop-by-hop beamforming over independent Rayleigh fading chan-nels is shown in [86]. The effect of antenna correlation on the performance ofMRC/MRC system has been investigated for Rayleigh fading in [87]. Theseworks [86,87] considered the variable-gain AF relay and Rayleigh fading chan-nels. For Nakagami-m fading channels, the variable-gain and fixed-gain AFrelay have been taken into account in [88] and [89], respectively. Finally, acomparison between these two variable-gain and fixed-gain relaying schemesfor MRC/MRC system has been conducted in [90].

One common aspect of the three mentioned MIMO AF relaying schemes,i.e., TAS/MRC, TAS/TAS, and MRC/MRC, is that the system requires fullCSI knowledge of the two hops to obtain the full diversity gain. Recently,it has been shown that by using orthogonal space-time block codes (OST-BCs) [92, 93] over dual-hop relay networks, one may achieve the full diver-sity order without acquiring such large amount of CSI knowledge [94–102].Specifically, the error rate performance of OSTBC transmission with fixed-gain AF relay over Rayleigh fading channels has been obtained in [95, 96].These works have been extended to study the impact of line-of-sight (LOS)in [94]. Here, the relay simply amplifies the received messages with a fixedvalue and forwards the resulting signals to the destination without any ad-ditional operation. This may cause noise accumulation at the destination.

Introduction 9

Alternatively, the relay may decouple the source messages using the squaringapproach [92] and then re-construct a new OSTBC before forwarding themto the destination. This relaying operation has been shown to provide fulldiversity gain without requiring the full CSI at the source as in the threeabove mentioned approaches, i.e., TAS/MRC, TAS/TAS, and MRC/MRC.As such, it has attracted numerous scholars’ interest [97,100–102]. By consid-ering a single-antenna relay, the error rate for OSTBC transmission over AFrelay networks with squaring approach has been derived for single-antennaand multiple-antenna destination in [98, 100], respectively. For Nakagami-mfading, the performance of OSTBCs transmission with squaring approach hasbeen derived for independent and correlated channels in [101, 102], respec-tively.

3.1 Realistic MIMO Environment: Antenna Correlation

and Rank-Deficiency

The benefit of multiple-antenna systems has been achieved by exploiting therich scattering nature of MIMO channels together with the assumption ofsufficient spacing between antenna elements so that the channels can undergoindependent fading. However, due to the space-limit at mobile terminals,the closely co-located antennas will induce spatial correlation between thesignals [103–106], which significantly degrades the diversity gain offered by theantenna arrays. In addition, the rank-deficiency effect of the channel matrix,namely, keyhole or pinhole as shown in Fig. 6, reduces the multiplexing gainsof MIMO systems [103, 107–114]. As such, to precisely assess the potentialof MIMO extension to cooperative communications for practical transmissionenvironment, these two important channel impairments should be taken intoaccount.

Recently, by considering a downlink communication with DF relay, theauthors in [115,116] have demonstrated that cooperative communications canmitigate the loss in multiplexing gain inflicted by the deleterious keyhole ef-fect. Motivated by these works, the diversity gain of MIMO AF relay hasbeen studied for keyhole and multi-keyhole in [117, 118], respectively. Theseworks are limited to the case where only the second hop is keyhole and therelay performs linear operation (fixed-gain AF relaying). It has been shownthat when the first hop is keyhole-free, the diversity order of min(nR, nD) canbe achieved, where nR and nD are the number of antennas at relay and desti-nation, respectively. A more general model where single-hop/dual-hop sufferskeyhole together with linear/squaring approach at the relay has been inten-

10 Introduction

TxRx

Keyhole

Figure 6: Keyhole effect in multiple-antennas systems.

sively studied in [119], which reveals important insights into the behavior ofsystem performance.

4 Cognitive Relay Networks

Frequency spectrum is one of the most important resources of wireless com-munication. However, it has been shown by many recent measurements thatonly a small portion of licensed radio spectrum is occupied at a certain time,see e.g., [120–127]. Cognitive radio, invented by Mitola [128], is a promisingsolution to alleviate the spectrum under-utilization, where the secondary user(SU) may be allowed to use the spectrum which is assigned prior to the pri-mary user (PU). Cognitive radio networks (CRNs) can be mainly classifiedas overlay, interweave, and underlay networks. In overlay CRN, both SU andPU occupy the spectrum at the same time and the SU utilizes the knowledgeof PU’s CSI to perform dirty paper coding so that the interference from PUis mitigated [129]. In contrast, in interweave CRN, the SU is allowed to usethe spectrum only when it is not occupied by the PU [130]. As such, thistechnique can be considered as opportunistic access. In an underlay network,however, the SU simultaneously occupies the spectrum with the PU as longas its interference on the primary network does not cause any harmful in-

Introduction 11

terference on the PU [131]. In our work, we are interested in the underlayCRN, also known as spectrum-sharing as shown in Fig. 7, since it requires theleast implementation complexity in comparison with overlay and interweaveapproaches.

PU

SU

Power Power

Frequency Frequency

Figure 7: Cognitive radio networks: a) Interweave, and b) underlay.

However, the signal-to-interference plus noise ratio (SINR) at the sec-ondary receiver of an underlay CRN is usually low due to: i) The limitation oftransmit powers of the secondary system, and ii) The interfering signals fromthe primary transmitter. Recently, cooperative communication has been ap-plied to underlay CRN to exploit the distributed spatial diversity gain undera limited transmit power condition. Generally, CRN with relaying has beenconsidered for DF relays [132–140] and AF relays [141–145]. In particular, theOP of cognitive DF relay networks has been first reported in [132]. Then, theSEP performance has been derived in [139]. The best DF relaying selectionhas been considered for CRN under spectrum sharing approach in [133,134].

In cognitive relay networks, due to the channel from the secondary sourceto the primary receiver, the end-to-end signal-to-noise ratio (SNR) at the sec-ondary destination is the combination of correlated RVs. This is because thereexists a common RV, i.e., channel gain of the link from secondary source to theprimary receiver, in each individual SNR corresponding to the transmissionover each relay. As a result, the end-to-end SNR is now given in the form of themaximum of multiple correlated RVs although the fading under considerationis independently distributed. This fact is also witnessed in previous works,e.g., [132, Eq. (6)], [134, Eq. (3)], [141, Eq. (6)], and [133, Eq. (9)]. How-ever, the statistical dependence among the RVs was not taken into account inthose works. As a result, the derivations presented in [132–134, 141] serve asa bound. In particular, they are lower bounds for the OP [132, 133, 141] and

12 Introduction

upper bounds for the ergodic capacity [134]. To overcome these drawbacksnoticed in previous works, an exact analysis has been introduced in [138] forcognitive DF relay systems. In this work, the authors have derived exactclosed-form expression for cognitive networks with best DF relay selection bytaking into account the statistical dependence effect among the RVs. For cog-nitive AF relay networks, an exact closed-form expression given in elementaryfunctions has been reported in [142]. Then, the spatial diversity technique isapplied for cooperative cognitive networks, where the selection combining isused at the destination to maximize the SNR [143]. In this work, the depen-dence among two RVs has been clearly taken into account, which results inan exact closed-form expression of the OP for the considered system.

Besides BRS, other important relaying schemes, e.g., partial relay selec-tion, have been also included in cognitive relay networks [141, 146]. Most ofthe previous works have only focused on Rayleigh fading channel for cogni-tive relay networks based on spectrum sharing. An extension to Nakagami-mfading channels for DF and AF relaying has been given in [137] and [145],respectively. However, all of these works have assumed that the primarytransmitter is located far away from secondary networks, and hence the ef-fect from primary transmitter can be neglected. Being one of the importantaspects of cognitive relay networks, the signals from the primary network, in-flicted by the concurrence of the transmission from the primary transmitter,may severely interfere the secondary receiver. It has been shown in [144] that,when the interference power from primary transmitter is proportional to thesecondary transmitter, it is useless to allow the secondary network to occupythe spectrum as the outage occurs for the whole considered SINR range.

5 Thesis Contribution

Although the performance of cooperative communications has been exten-sively studied in the research community, most of these works have assumedperfect conditions. Hence, this thesis has aimed at investigating coopera-tive communications with practical constraints such as antenna correlation,keyhole/multi-keyhole effects, CCI, and interference power constraint.

This thesis consists of six parts. Part I analyzes the performance ofMIMO dual-hop AF relay systems using OSTBCs over arbitrarily correlatedNakagami-m fading channels. Part II investigates the impact of the keyholeeffect of the MIMO channel matrix, where the relay terminal can deploy linearand squaring approaches. Part III extends the keyhole effect to a more gen-

Introduction 13

eral case, namely, multi-keyhole, where the rank of the channel matrix can bevaried from one to full-rank. In Part IV, we propose a DSTC for the two-wayrelay to compensate the loss in spectral efficiency of its one-way counterpartwhile keeping the full diversity gain. In Part V, the joint impact of CCI andantenna correlation is taken into account for a beamforming two-hop AF relaynetwork. Finally, Part VI investigates the performance of spectrum sharingAF relay networks in an interference-limited environment, where the interfer-ence induced by the transmission of primary networks and the existence of adirect link are taken into account. In the following, the contribution of eachpart is summarized.

5.1 PART I: Orthogonal Space-Time Block Codes with

CSI-Assisted Amplify-and-Forward Relaying in Cor-

related Nakagami-m Fading Channels

In this part, the performance of MIMO dual-hop AF relay systems with OS-TBCs transmission over arbitrarily correlated Nakagami-m fading channels isanalyzed. Below are important contributions of this part:

• Closed-form expressions for the end-to-end OP and the SEP with arbi-trary number of transceiver antennas and general correlation matricesare derived. Their mathematically tractable forms readily enable usto evaluate the performance of MIMO AF relay systems that utilizesOSTBCs.

• For sufficiently high signal-to-noise ratios, asymptotically tight approx-imations for the OP and SEP are also attained which reveal insightsinto the effects of fading parameters and antenna correlation on systemperformance.

• We prove that for a full-rank correlation matrix, the antenna correlationhas no impact on the achievable diversity gain which is equal to theminimum of the sum of fading parameters between the two hops.

5.2 PART II: Keyhole Effect in Dual-Hop MIMO AF

Relay Transmission with Space-Time Block Codes

This part extends the work in Part I, where the channel is assumed to be offull-rank, to the case of rank-deficiency. In particular, we analyze the perfor-mance of a downlink communication system where the amplifying processing

14 Introduction

at the relay can be implemented by either the linear or squaring approach.Our contributions in this part are summarized as follows:

• We derive tractable asymptotic SEP expressions, which enable us toobtain both diversity and array gains. An important observation cor-roborated by our studies is that for satisfying the tradeoff between per-formance and complexity, we should use the linear approach for singlekeyhole effect (SKE) and the squaring approach for double keyhole effect(DKE).

• Our finding reveals that for the downlink system, i.e., nS > min(nR, nD),the linear approach can provide the full achievable diversity gain ofmin(nR, nD) with SKE, where nS, nR, and nD are the number of antennasat source, relay, and destination, respectively. However, for the case thatboth the source-relay and relay-destination links experience the keyholeeffect, i.e., DKE, the achievable diversity order is only one regardless ofthe number of antennas. In contrast, utilizing the squaring approach,the overall diversity gain can be achieved as min(nR, nD) for both SKEand DKE.

5.3 PART III: Multi-Keyhole Effect in MIMO AF Re-

lay Downlink Transmission with Space-Time Block

Codes

This part generalizes the keyhole effect considered in Part II to the multi-keyhole scenario. Specifically, we study the impact of multi-keyhole, i.e.,the bridge between single keyhole and full-scattering MIMO channels, on theperformance of MIMO AF relay downlink transmission with OSTBCs. Thecontributions of this part are summarized as follows:

• We derive an analytical SEP expression for the considered system witharbitrary number of keyholes.

• Moreover, SEP approximations in the high SNR regime for several im-portant special scenarios of multi-keyhole channels are further derived.These asymptotic results provide important insights into the impact ofsystem parameters on the SEP performance.

Introduction 15

5.4 PART IV: Distributed Space-Time Coding in Two-

Way Fixed Gain Relay Networks over Nakagami-m

Fading

One-way relay networks have a loss in spectral efficiency due to the multi-hoptransmission. In this part, to remedy this shortcoming, the fixed gain AFrelaying has been applied to distributed-Alamouti space-time code (DASTC)two-way transmission. Below are the contributions of this part:

• Closed-form expressions for ergodic sum-rate and pairwise error prob-ability (PWEP) over Nakagami-m have been derived. The two-waysystem has been shown to surpass the commonly considered one-waycounterpart by one nats/s/Hz, which is a remarkable improvement inspectral efficiency knowing that the contemporary wireless system cansupport up to 2-3 nats/s/Hz.

• The final results are given in the form of Fox-H function which readilyenable us to evaluate the PWEP and capacity of the proposed schemein some representative scenarios. Moreover, the asymptotic PWEP re-vealing the array and diversity gains are also derived.

5.5 PART V: Beamforming in Two-Way Fixed Gain

Amplify-and-Forward Relay Systems with CCI

In this part, we analyze the outage performance of a two-way fixed gain AFrelay system with beamforming, arbitrary antenna correlation, and CCI. Ourcontributions in this part are as follows:

• Assuming CCI at the relay, we derive the exact individual user OP inclosed-form. Additionally, we also investigate the system OP of theconsidered network, which is declared if any of the two users is in trans-mission outage.

• Our results indicate that in this system, the position of the relay playsan important role in determining the user OP as well as the system OPvia such parameters as signal-to-noise imbalance between the two hops,antenna configuration, spatial correlation, and CCI power.

• To render further insights into the effect of antenna correlation and CCIon the diversity and array gains, an asymptotic expression which tightlyconverges to exact results is also derived.

16 Introduction

5.6 PART VI: Cognitive Cooperative Communication

with Amplify-and-Forward Relay and Spectrum-

Sharing Approach

Radio frequency spectrum is a scarce and expensive resource of wireless net-works. However, most of the licensed spectrum bands are not efficiently uti-lized. CRN has been proposed as a practical technique to improve spectrumutilization. The performance of CRN may be limited since the transmit powerat the SU is strictly governed by the interference power constraint imposed bythe PU. As a result, utilizing cooperative communication is essential to im-prove CRN performance. In this part, we investigate the advantage of usingrelaying in CRNs. Our contributions in this part are as follows:

• The exact closed-form expression for the OP of cognitive radio dual-hop AF relay networks is derived. The tractable expression of the OP,given in the form of elementary functions, readily enables us to evaluatethe effect of the PU on the secondary system performance. It has beenshown that the use of AF relaying significantly improves the performanceof CRNs compared to the direct transmission.

• The spatial diversity for cognitive dual-hop relay networks is analyzedunder interference power constraint imposed by the PU. The tractableclosed-form OP expression readily enables us to evaluate the systemperformance, which indicates the significance of using diversity combin-ing in a distributed fashion in CRNs with underlay spectrum sharingapproach.

• The effect of the primary network on spectrum sharing AF relaying isalso taken into account. We show that under fixed interference fromthe primary network, the diversity order of the secondary network isnot affected but only the array gain. However, when the interferencepower is dependent on the average SNR of the secondary network, it isinfeasible to operate the secondary system as an irreducible error floorexists for the whole SNR regime.

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Introduction 25

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Part I

Part I

Orthogonal Space-Time Block Codes with

CSI-Assisted Amplify-and-Forward Relaying in

Correlated Nakagami-m Fading Channels

Part I is published as

T. Q. Duong, G. C. Alexandropoulos, T. A. Tsiftsis, and H.-J. Zepernick, “Or-thogonal Space-Time Block Codes with CSI-Assisted Amplify-and-ForwardRelaying in Correlated Nakagami-m Fading Channels,” IEEE Trans. on Veh.

Techno., vol. 60, no. 3, pp. 882–889, Mar. 2011.

Based on

T. Q. Duong, H.-J. Zepernick, T. A. Tsiftsis, and V. N. Q. Bao, “Amplify-and-forward MIMO relaying with OSTBC over Nakagami-m fading channels,” inProc. IEEE International Communications Conference, Cape Town, SouthAfrica, May 2010.

T. Q. Duong, G. C. Alexandropoulos, T. A. Tsiftsis, and H.-J. Zepernick,“Outage Probability of MIMO AF Relay Networks over Nakagami-m FadingChannels,” Electron. Lett., vol. 46, no. 17, pp. 1229–1231, Sep. 2010.

Orthogonal Space-Time Block Codes with

CSI-Assisted Amplify-and-Forward Relaying in

Correlated Nakagami-m Fading Channels

Trung Q. Duong, George C. Alexandropoulos,

Hans-Jurgen Zepernick, and Theodoros A. Tsiftsis

Abstract

In this paper, the performance of multiple-input multiple-output(MIMO) dual-hop amplify-and-forward (AF) relay systems using or-thogonal space-time block codes (OSTBCs) over arbitrarily correlatedNakagami-m fading channels is analyzed. In particular, closed-form ex-pressions for the end-to-end outage probability (OP) and the symbolerror probability (SEP) with arbitrary number of transceiver anten-nas and general correlation matrices are derived. Their mathematicallytractable forms readily enable us to evaluate the performance of MIMOAF relay systems that utilize OSTBCs. For sufficiently high signal-to-noise ratios, asymptotically tight approximations for the OP and SEPare also attained which reveal insights into the effects of fading parame-ters and antenna correlation on the system’s performance. Furthermore,we prove that the correlation has no impact on the achievable diversitygain which is equal to the minimum of the sum of fading parametersbetween the two hops. Selected numerically evaluated results are pre-sented showing an excellent agreement between the proposed analysisand equivalent Monte-Carlo simulations.

1 Introduction

As one of the prominent space-time processing techniques, orthogonal space-time block codes (OSTBCs) are known to provide full spatial diversity gaintogether with low encoding/decoding complexity [1]. For some practical ap-plications, e.g., cellular communications in rural areas where the population

37

38 Part I

density is low, building a whole conventional cellular system is not econom-ically beneficial. In such scenarios, relay/cooperative communication is analternative method to obtain multiple-input multiple-output (MIMO) bene-fits in a distributed fashion [2, 3].

The transmission of OSTBCs in amplify-and-forward (AF) relay networkshas lately gained great attention in the research community (see, e.g., [4,5, 6, 7, 8] and the references therein). The symbol error probability (SEP)of MIMO systems, where the source deploys OSTBCs to communicate withthe destination through the assistance of semi-blind AF relays, has been in-vestigated over Rayleigh fading channels [5]. Specifically, the average SEPhas been expressed in the form of an integral whose integrand contains thehypergeometric function. By taking into account the line-of-sight effect, theanalysis for Rayleigh-Rayleigh fading in [5] has been extended in dual-hopRayleigh-Rician fading [6]. Recently, an exact expression for the bit errorrate (BER) of binary phase shift keying modulation (BPSK) for OSTBCswith semi-blind AF relays over Rayleigh fading has been presented in [7]. Forthe same fading conditions, in [4], the BER of MIMO systems that utilizeOSTBCs and an AF relay that possesses channel state information (CSI) hasbeen studied. Taking the direct link into consideration, the SEP and the out-age probability (OP) of MIMO CSI-assisted AF relaying with OSTBCs overRayleigh fading has been analyzed in [8]. Very recently, the performance ofMIMO AF relay systems with OSTBCs transmission has been investigatedfor independent Nakagami-m fading channels [9].

In this paper, we analyze the performance of CSI-assisted dual-hop AF re-laying with OSTBCs over arbitrarily correlated and not necessarily identicallydistributed (c.n.i.d.) Nakagami-m fading channels. Analytical expressions forthe end-to-end OP and SEP are obtained in compact forms readily enabling usto investigate the system’s performance. Our analytical derivations are validfor integer values of the fading severity parameters and numerically evalu-ated performance results perfectly agree with the equivalent results obtainedby means of Monte-Carlo simulations. Furthermore, to provide insights onhow fading parameters and antenna correlation affect performance, we de-rive asymptotic expressions for the OP and SEP which reveal the diversitygain of MIMO AF relay systems with OSTBCs. It has been observed thatthe asymptotic results definitely converge to the exact values for sufficientlyhigh signal-to-noise ratios (SNRs). Moreover, we prove that the correlationdegrades the performance but not the achievable diversity gain which strictlydepends on the fading parameters. In particular, the diversity gain is equalto the minimum of the sum of the fading parameters of the two hops which

OSTBCs with CSI-Assisted AF Relaying in Correlated Nakagami-m Fading 39

implies that, between the two hops, the more severe link solely determines thediversity gain.

The remaining of this paper is organized as follows. In Section 2, we brieflyreview the system and channel model for cooperative communications with AFrelay and OSTBCs over c.n.i.d. Nakagami-m fading channels. In Section 3, wederive closed-form expressions for the OP and SEP. The asymptotic analysis ispresented in Section 4 to reveal the diversity gain of MIMO AF relay systemsusing OSTBCs. Numerical results are provided in Section 5 to validate ouranalysis. Finally, Section 6 concludes the paper.

Notation: Throughout the paper, boldface upper and lower case lettersrepresent matrices and vectors, respectively, whereas IIIn represents the n× nidentity matrix; superscripts ∗, T, and † stand for complex conjugate, trans-pose, and transpose conjugate, respectively; ‖y‖F denotes the Frobenius normof yyy and |x| indicates the envelope of x; U(x) is the unit step function [10,p. xliv], whereas E · is the expectation operator; Γ (·) is the Gamma func-tion [10, eq. (8.310.1)], Γ (·, ·) is the incomplete Gamma function [10, eq.(8.350.2)], and Kn (·) is the nth-order modified Bessel function of the secondkind [10, eq. (8.432.3)].

2 System and Channel Model

Let us consider a dual-hop relay-based wireless system with an L1-antennasource S, a single-antenna relay R, and an L2-antenna destination D. All chan-nels are assumed to undergo the quasi-static Nakagami-m fading, i.e., fadingcoefficients remain constant for the duration of the coherence time (consist-ing of multiple intervals of block-length of an OSTBC codeword) and changeindependently for every coherence time. All terminals are assumed to operatein half-duplex mode, i.e., they cannot transmit and receive simultaneously.In the first hop, S transmits an L1 × K OSTBC codeword matrix XXX whichconveys N symbols, x1, x2, . . . , xN , selected from a signal constellation withtransmit power per symbol Ps. Since N symbols are transmitted in totalwithin a block of K symbols, the code rate of the OSTBC is Rc = N/K.During the first hop, the 1 ×K signal vector yyy1 received at R is given by

yyy1 = hhh1XXX + zzz1 (1)

where hhh1 = [h11, h12

, . . . , h1L1] ∈ C1×L1 consists of the possibly correlated

channel coefficients for the S → R link and zzz1 is a 1×K complex additive white

40 Part I

Gaussian noise (AWGN) vector at R whose elements are complex Gaussianrandom variables (RVs) with zero mean and variance N0.

In the second hop, R amplifies its received signal from S with a scalaramplifying gain G and forwards the resulting signal to D with the same powerPs consumed at S during the first hop. The L2×K signal matrix YYY 2 receivedat D can be expressed as

YYY 2 = hhhT2Gyyy1 +ZZZ2 (2)

where hhh2 = [h21, h22

, . . . , h2L2] ∈ C1×L2 comprises of the possibly correlated

channel coefficients for the R → D links and ZZZ2 is an L2 × K AWGN ma-trix whose elements are complex Gaussian RVs with zero mean and varianceN0. Assuming that R has full knowledge of hhh1, G can be determined underthe constraint that S and R consume the same amount of power, yielding

G2 =(

‖hhh1‖2F

)−1

where γ0 = Ps/N0 is the average transmit SNR. At D,

due to the orthogonal property of OSTBCs, the maximum-likelihood decod-ing becomes the symbol-wise decoding, i.e., each transmitted symbol xn, withn = 1, 2, . . . , N , can be decoded independently. Hence, similar to [6, 5, 7, 4],the instantaneous SNR at D can be accurately approximated by

γOSTBC =γ0

L1Rc

‖hhh1‖2F ‖hhh2‖2

F

‖hhh1‖2F + ‖hhh2‖2

F

= γγ1γ2

γ1 + γ2(3)

where γ ,γ0

L1Rc, γ1 ,

∑L1

ℓ=1 |h1ℓ|2, and γ2 ,

∑L2

ℓ=1 |h2ℓ|2.

3 End-to-End Performance Analysis

In this section, we derive the OP and the SEP performance of dual-hop AFrelay systems with OSTBCs over c.n.i.d. Nakagami-m fading channels. Webegin our analysis by introducing some preliminaries on the closed-form statis-tics for the sum of Gamma RVs.

3.1 Statistical Background on the Sum of Gamma RVs

Let hAℓ, A ∈ 1, 2 and ℓ = 1, 2, . . . , LA, be a Nakagami-m faded channel

coefficient with fading severity parameter mAℓand average fading power ΩAℓ

.Clearly, |hAℓ

|2 is a Gamma distributed RV with parameter set (mAℓ, αAℓ

)


where αAℓ=

mAℓ

ΩAℓ

. For the most general case of c.n.i.d. channels, |hAℓ|2’s

are arbitrarily correlated with power correlation matrix (CM) Σ ∈ RLA×LA .This matrix is symmetric, positive definite, and given by Σk,ℓ , 1 for k = ℓ,

with k = 1, 2, . . . , LA, and Σk,ℓ , ρk,ℓ for k 6= ℓ, where ρk,ℓ ∈ [0, 1) is thecorrelation coefficient between |hAk

|2 and |hAℓ|2 [11, eq. (9.195)].

To obtain the statistics of the sum of c.n.i.d. Gamma RVs, γA =∑LA

ℓ=1 |hAℓ|2,

with αAℓ6= αAk

∀ℓ 6= k1, γA can be transformed to a sum of independentGamma RVs [13]. Specifically, γA can be expressed as the sum of κA in-

dependent Gamma RVs with parameter sets(

µAℓ

2 ,µAℓ

4λAℓ

)

∀ ℓ = 1, 2, . . . , κA.

Hereinafter, λ , [λA1, λA2

, . . . , λAκA] ∈ R

1×κA

+ consists of the κA distinct

eigenvalues of matrix C ∈ RNA×NA where NA = 2∑LA

ℓ=1 mAℓ. Matrix C is

obtained as C = bbT, where b ∈ RNA×1 is an auxiliary vector defined as b ,

[bT1 ,b

T2 , . . . ,b

TLA

]T with bℓ ∈ R2mAℓ

×1 given by bℓ , [bℓ,1, bℓ,2, . . . , bℓ,2mAℓ]T.

The elements of bℓ’s are given by

bk,j bℓ,n =

ΩAk

2mAk

, if k = ℓ and j = n

ρk,ℓ

2

√

ΩAkΩAℓ

mAkmAℓ

, if k 6= ℓ and

j = n ≤ 2 min mAk,mAℓ

0, otherwise

(4)

where k = 1, 2, . . . , LA. Moreover, let us define µ , [µA1, µA2

, . . . , µAκA] ∈

Z1×κA

+ as the corresponding algebraic multiplicities of the κA distinct eigen-

values given by λ such that∑κA

p=1 µAp= 2

∑LA

ℓ=1 mAℓ. The probability den-

sity function (PDF) and cumulative distribution function (CDF) of γA areexpressed, respectively, as [13]

fγA(γ) =

κA∑

ℓ=1

χAℓ∑

k=1

ΞA (ℓ, k) (βAℓ)kγk−1 exp (−βAℓ

γ) (5)

FγA(γ) =

κA∑

ℓ=1

χAℓ∑

k=1

ΞA (ℓ, k)

[

1 − Γ (k, βAℓγ)

Γ (k)

]

(6)

where βAℓ=

µAℓ

4λAℓ

, χAℓ=

µAℓ

2 , and the weighting coefficients ΞA (ℓ, k) are given

by [13, eq. (7)].

1Following the analysis presented in [12], analytical expressions for the statistics of γA

for the case of identical fading parameters, i.e., αAℓ= αAk

∀ ℓ 6= k, can be easily derived.

42 Part I

3.2 End-to-End Performance of AF Relaying with OST-

BCs over Nakagami-m Channels

As observed from (3), to analyze the performance of the systems under con-sideration, the statistical properties of γ3 = γ1γ2

γ1+γ2need to be derived. The

following theorem will be helpful to obtain exact closed-form expressions forthe OP and SEP of MIMO AF relay systems using OSTBCs over c.n.i.d.Nakagami-m fading channels.

Theorem 1 The CDF of γ3 is given by

Fγ3 (γ) =

κ2∑

ℓ=1

χ2ℓ∑

k=1

Ξ2 (ℓ, k)

[

1 − Γ (k, β2ℓγ)

Γ(k)

]

+

κ1∑

ℓ=1

χ1ℓ∑

k=1

Ξ1 (ℓ, k)

κ2∑

u=1

χ2u∑

v=1

Ξ2 (u, v)

Γ (v)

× (β2u)v

[v−1∑

j=0

(v − 1)!

(v − 1 − j)!(β2u

)−j−1

γv−j−1 exp (−β2uγ)

−k−1∑

i=0

(β1ℓ)i

i!

v+i−1∑

w=0

2

(

v + i− 1

w

)

ξν2

ℓ,uγv+i exp (−ηℓ,uγ)Kν(ζℓ,uγ)

]

. (7)

Proof The proof is given in the Appendix A.

Using Theorem 1, we hereafter derive exact closed-form expressions for theOP and the SEP.

3.2.1 Outage Probability (OP)

The OP is defined as the probability that the instantaneous SNR falls be-low a given threshold γth. Using (7), the OP of dual-hop AF relaying withOSTBCs over c.n.i.d. Nakagami-m fading channels is easily obtained as

Pout = Fγ3

(

γthγ

)

.

3.2.2 Symbol Error Probability (SEP)

We first start by making use of a common result presented in [14], in whichthe SEP can be directly evaluated from the CDF of γOSTBC as

Perr =a√b

2√π

∞∫

0

Fγ3

(

t

γ

)

e−bt√tdt (8)


where a and b are constants that depend on the specific modulation scheme.Substituting (7) in (8), the SEP can be expressed as a sum of two integralsI1 and I2, i.e., Perr = I1 + I2, where I1 and I2 are given, respectively, by

I1 =a√b

2√π

κ2∑

ℓ=1

χ2ℓ∑

k=1

Ξ2 (ℓ, k)

∞∫

0

[

1 − Γ

(

k,β2ℓ

t

γ

)

/Γ(k)

]

e−bt√tdt (9)

and

I2 =a√b

2√π

κ1∑

ℓ=1

χ1ℓ∑

k=1

Ξ1 (ℓ, k)

κ2∑

u=1

χ2u∑

v=1

Ξ2 (u, v) (β2u)v

Γ (v)(I3 − I4) . (10)

It is noted that (9) follows immediately from the binomial theorem togetherwith the help of [10, eq. (8.312.2)]. Furthermore, I3 and I4 in (10) are thetwo following tabulated integrals

I3 =

v−1∑

j=0

(v − 1)!

(v − 1 − j)!(β2u

)−j−1

γj+1−v∞∫

0

tv−j−32 e

−(

β2uγ

+b)

tdt, (11)

I4 =

k−1∑

i=0

βi1ℓ

i!

v+i−1∑

w=0

2

(

v + i− 1

w

)

ξν2

ℓ,uγ−v−i

∞∫

0

tv+i−1/2e−(ηℓ,u

γ+b)tKν

(

ζℓ,ut

γ

)

dt.

(12)

Next, using [10, eq. (8.312.2)] for obtaining I3 and applying [10, eq. (6.621.3)]for I4, the SEP of dual-hop AF relaying with OSTBCs over c.n.i.d. Nakagami-m fading channels is derived as

Perr =a√b

2√π

κ2∑

ℓ=1

χ2ℓ∑

k=1

Ξ2 (ℓ, k)

[√

π

b−k−1∑

i=0

(

β2ℓ

γ

)i(β2ℓ

γ+ b

)−i− 12 Γ(i+ 1/2)

i!

]

+a√b

2√π

κ1∑

ℓ=1

χ1ℓ∑

k=1

Ξ1 (ℓ, k)

κ2∑

u=1

χ2u∑

v=1

Ξ2 (u, v) (β2u)v

Γ (v)

×[v−1∑

j=0

(v−1)!(v−1−j)! (β2u

)−j−1 γj+1−vΓ(

v − j − 12

)

(

β2u

γ+ b

)j+12−v

−k−1∑

i=0

(β1ℓ)i

i!

v+i−1∑

w=0

2

(

v + i− 1

w

)

ξν2

ℓ,uγ−v−i2F1

(

µ+ ν, ν + 12 ;µ+ 1

2 ;σ − τ

σ + τ

)]

.

(13)

44 Part I

In (13), µ = v + i+ 12 , σ =

ηℓ,u

γ + b, τ =ζℓ,u

γ , =√π(2τ)ν

(σ+τ)µ+ν

Γ(µ+ν)Γ(µ−ν)Γ(µ+1/2) , and

2F1 (·, ·; ·; ·) is the Gauss hypergeometric function defined in [15, eq. (2.12.1)].

4 Asymptotic Analysis for the High SNR Re-

gime

To render insights into the effect of fading parameters within the two hopsas well as antennas’ correlation on the system’s performance, asymptoticallytight approximations for the OP and SEP are derived in this section. In thisregard, utilizing a similar treatment to [16, 9], we introduce the followinglemmas.

Lemma 1 The PDF of γA, with A ∈ 1, 2, can be approximated as

fγA(γ)

γ→0≈ cAγdA (14)

where dA = −1 +∑κA

i=1 χAiand cA =

∏κA

i=1(χAi/βAi)

χAi

(dA)! .

Proof Since γA is the sum of κA independent Gamma RVs, recursively ap-

plying [16, Lemma 1] yields (14).

Lemma 2 The PDF of γ3 can be approximated as

fγ3 (γ)γ→0≈ c3γ

d3 (15)

where

d3 = −1 + min

(

κ1∑

ℓ=1

χ1ℓ,

κ2∑

ℓ=1

χ2ℓ

)

and

c3 =

∏κ1ℓ=1

β1ℓ

(−1+∑κ1

ℓ=1χ1ℓ

)!if∑κ1

ℓ=1 χ1ℓ<∑κ2

ℓ=1 χ2ℓ∏κ1

ℓ=1β1ℓ

(−1+∑κ1

ℓ=1χ1ℓ

)!+

∏κ2ℓ=1

β2ℓ

(−1+∑κ2

ℓ=1χ2ℓ

)!if∑κ1

ℓ=1 χ1ℓ=∑κ2

ℓ=1 χ2ℓ∏κ2

ℓ=1β2ℓ

(−1+∑κ2

ℓ=1χ2ℓ

)!if∑κ1

ℓ=1 χ1ℓ>∑κ2

ℓ=1 χ2ℓ

Proof See Appendix B.


5 10 15 20 25 30

10-6

10-5

10-4

10-3

10-2

10-1

100

th = 10 dB

= 0.4L

1 = L

2 = 3

Case 1

Case 3

Case 2

correlated Nakagami-m fading

Analysis (closed-form) Analysis (approximation) Simulation

SNR (dB)

End-

to-E

nd O

utag

e Pr

obab

ility

Figure 1: End-to-end OP of MIMO dual-hop AF relay systems with out-age threshold γth = 10 dB versus the average received SNR over c.n.i.d.Nakagami-m fading channels.

Proposition 1 The OP and SEP of dual-hop AF relaying with OSTBCs over

c.n.i.d. Nakagami-m fading channels can be asymptotically approximated,

respectively, as

Pout ≈c3 (L1Rcγth)

d3+1

d3 + 1γ−d3−10 , (16)

Perr ≈c3∏d3+1i=1 (2i− 1)

(d3 + 1)

(

L1Rc

2g

)d3+1

γ−d3−10 (17)

where g = sin2(

πM

)

.

Proof The proof is provided in the Appendix C.

The above proposition reveals the diversity gain of OSTBCs in AF relay sys-tems as

D(out) = D(err) =1

2min

(

κ1∑

ℓ=1

µ1ℓ,

κ2∑

ℓ=1

µ2ℓ

)

= min

(

L1∑

k=1

m1k,

L2∑

k=1

m2k

)

,

46 Part I

which exactly agrees with the diversity gain obtained in the case of indepen-dent Nakagami-m fading [9, eq. (23)]. Interestingly, the obtained diversitygain is strictly determined by the more severe faded hop.

5 Numerical Results and Discussion

In this section, numerically evaluated results for MIMO dual-hop AF relay sys-tems with OSTBCs over c.n.i.d. Nakagami-m fading channels are presented.Monte-Carlo simulations are conducted in order to verify the correctness ofthe derived closed-form expressions as well as the tightness of the presentedasymptotic approximations.

5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

10-1

100

Case 2

Case 1

Case 3

= 0.4L

1 = L

2 = 3

correlated Nakagami-m fading

Analysis (closed-form) Analysis (approximation) Simulation

SNR (dB)

Sym

bol E

rror

Pro

babi

lity

Figure 2: SEP of MIMO dual-hop AF relay systems for 8-PSK modulationversus the average received SNR over c.n.i.d. Nakagami-m fading channels.

The OP and SEP performance versus the average SNR γ of MIMO AFrelay systems with L1 = L2 = 3 antennas operating over various Nakagami-mfading channels are depicted in Figs. 1 and 2, respectively. In particular, thefollowing three different cases are considered for the fading severity parame-ters: 1) Case 1: The fading of the first hop is more severe than that of thesecond hop: m1ℓ

3ℓ=1 = 1, 1, 2 and m2ℓ

3ℓ=1 = 1, 2, 2; 2) Case 2: The


5 10 15 20 25 3010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

L1 = L

2 =3

th = 10 dB

= 0 = 0.2 = 0.4 = 0.6

SNR (dB)

End-

to-E

nd O

utag

e Pr

obab

ility

Figure 3: End-to-end OP of MIMO dual-hop AF relay systems with out-age threshold γth = 10 dB versus the average received SNR over c.n.i.d.Nakagami-m fading for different correlation coefficients.

severities of the two hops are the same: m1ℓ3ℓ=1 = m2ℓ

3ℓ=1 = 1, 1, 1; 3)

Case 3: The severity of the second hop is more rigorous than that of the firsthop: m1ℓ

3ℓ=1 = 1, 2, 3 and m2ℓ

3ℓ=1 = 1, 2, 2. In these figures, the ex-

ponentially decaying power model ΩAℓ= ΩA1

exp[−0.5(ℓ−1)] where A ∈ 1, 2with ℓ = 2 and 3, Ω11

= 0.74, and Ω21= 0.82 is assumed. Moreover, for the

CMs of S and D, the correlation model resulting from equispaced diversityantennas is used, i.e., the exponential correlation coefficient ρi,j among thei-th and j-th input channel is given by ρi,j = ρ|i−j| with i, j = 1, 2, and 3.

As it can clearly be seen from these two figures for ρ = 0.4, the analyticalresults for the OP with γth = 10 dB and SEP of 8-PSK modulation perfectlymatch with simulations. Moreover, the asymptotic approximations, revealingthe obtained diversity gains, are shown to be very tight to the exact valuesin the high SNR regime. As expected, a substantial improvement in the OPand the SEP performance can be obtained for Case 3 compared to the others.This happens because Case 3 owns the largest value of the minimum of thesum of fading severities between the two hops among the three considered

48 Part I

5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

L1 = L

2 =3

= 0 = 0.2 = 0.4 = 0.6

8-PSK

SNR (dB)

Sy

mbo

l Err

or P

roba

bilit

y

Figure 4: SEP of MIMO dual-hop AF relay systems for 8-PSK modulationversus the average received SNR over c.n.i.d. Nakagami-m fading for differentcorrelation coefficients.

cases.

The effect of antenna correlation on the OP and SEP performance ofMIMO AF relay systems over Nakagami-m fading is illustrated in Figs. 3and 4, respectively, for different values of γ. In these figures, the number ofantennas at S and D are set to L1 = L2 = 3 and the fading parameters of thetwo hops are given by m1ℓ

3ℓ=1 = 1, 2, 3 and m2ℓ

3ℓ=1 = 1, 2, 4. Similar

to the cases shown in Figs. 1 and 2, the correlation coefficient of both theCMs of S and D is designated as ρi,j = ρ|i−j| and the channel mean powerfor the two hops is ΩAℓ

= ΩA1exp[−0.5(ℓ− 1)], where i, j = 1, 2 and 3, ℓ = 2

and 3, and ΩA1= 1. As observed from Fig. 3 for γth = 10 and Fig. 4 for 8-

PSK modulation, respectively, antenna correlation degrades the OP and SEPperformance. However, it has no impact on the diversity gain. In fact, in thehigh SNR regime, all OP and all SEP curves are parallel, i.e., they have thesame slope, but are shifted to the right as ρ increases from 0 to 0.6.

To further investigate the effect of antenna correlation on the diversitygain of MIMO AF relay systems with OSTBCs over c.n.i.d. Nakagami-mfading channels, we study the OP and SEP performance for the following three


scenarios: 1) (L1, L2) = (2, 2) and mAℓ2ℓ=1 = 2, 3; 2) (L1, L2) = (3, 3) and

mAℓ3ℓ=1 = 1, 2, 2; and 3) (L1, L2) = (4, 4) and mAℓ

4ℓ=1 = 1, 1, 1, 2.

These scenarios are considered in Figs. 5 and 6 where the OP with γth = 10dB and the SEP of 8-PSK modulation are plotted versus SNR, respectively.The correlation coefficient is ρ = 0.5 in these figures and all other parametersare set the same as in Figs. 1 and 2.

As it is shown, all three scenarios yield the same diversity gain for boththe OP and SEP. Moreover, as expected, the best OP and SEP performanceis obtained when L1 = L2 = 4, i.e., when S and D have the largest numberof antennas among three cases. It is interesting to see that the diversity gaindoes not depend on the number of transmit and receive antennas as long as

the underlying systems have the same value of min(

∑L1

ℓ=1m1ℓ,∑L2

ℓ=1m2ℓ

)

.

5 10 15 20 25 30

10-6

10-5

10-4

10-3

10-2

10-1

100

(L1,L

2) = (2,2)

(L1,L

2) = (3,3)

(L1,L

2) = (4,4)

th = 10 dB

= 0.5

SNR (dB)

End-

to-E

nd O

utag

e Pr

obab

ility

Figure 5: End-to-End OP of MIMO dual-hop AF relay systems with outagethreshold γth = 10 dB and correlation coefficient ρ = 0.5 versus the aver-age received SNR over c.n.i.d. Nakagami-m fading for different number oftransceiver’s antennas.

50 Part I

5 10 15 20 25 30

10-6

10-5

10-4

10-3

10-2

10-1

100

(L1,L

2) = (2,2)

(L1,L

2) = (3,3)

(L1,L

2) = (4,4)

8-PSK

SNR (dB)

Sym

bol E

rror

Pro

babi

lity

Figure 6: SEP of MIMO dual-hop AF relay systems for 8-PSK modulation andcorrelation coefficient ρ = 0.5 versus the average received SNR over c.n.i.d.Nakagami-m fading for different number of transceiver’s antennas.

6 Conclusions

In this paper, we have analyzed the performance of MIMO dual-hop AF re-lay systems with OSTBCs transmission in arbitrarily correlated Nakagami-mfading channels. In particular, closed-form expressions for the OP and theSEP have been derived. The obtained formulas have been expressed in multi-fold summations of Bessel and Gauss hypergeometric functions. Furthermore,to gain insights into the diversity gain offered by such systems, we have alsoderived asymptotic approximations for the OP and SEP which were shownto be very tight in the high SNR regime. It has been proved that antennacorrelation has no impact on the achievable diversity gain which is equal tothe minimum of the sum of fading severity parameters between the two hops.Selected numerically evaluated performance results have exhibited a goodagreement with equivalent Monte-Carlo simulations, as such, validating theaccuracy of the presented analysis. All results have clearly shown the impactof fading severity parameters and antenna correlation on the OP and SEPperformance of the systems under consideration.


Appendix A: Proof of Theorem 1

The CDF of γ3 = γ1γ2γ1+γ2

can be derived as Fγ3 (γ) = Fγ2 (γ) + J where

J =

∫ ∞

γ

κ1∑

ℓ=1

χ1ℓ∑

k=1

Ξ1 (ℓ, k)

1 −Γ(k,

β1ℓγγ2

γ2−γ )

Γ(k)

×κ2∑

u=1

χ2u∑

v=1

Ξ2 (u, v)βv2u

Γ (v)γv−12 exp (−β2u

γ2) dγ2. (18)

To simplify J , we further presume that χ1ℓ, χ2u

∈ Z+ ∀ ℓ, u. For this case, γ(·)can be expressed as a finite sum [10, eq. (8.352.2)]. Applying the binomialtheorem [10, eq. (1.111)] in (18), making use of [10, eq. (3.381.4)], [10, eq.(3.471.9)], and after some algebraic manipulations, yields

J =

κ1∑

ℓ=1

χ1ℓ∑

k=1

Ξ1 (ℓ, k)

κ2∑

u=1

χ2u∑

v=1

Ξ2 (u, v) (β2u)v

Γ (v)

[v−1∑

j=0

(v − 1)!

(v − 1 − j)!(β2u

)−j−1

× γv−j−1e(−β2uγ) −k−1∑

i=0

(β1ℓ)i

i!

v+i−1∑

w=0

2

(

v + i− 1

w

)

ξν2

ℓ,uγv+ie(−ηℓ,uγ)Kν(ζℓ,uγ)

]

.

(19)

In (19), ν = w − i + 1, ξℓ,u =β1ℓ

β2u, ηℓ,u = β1ℓ

+ β2u, and ζℓ,u = 2

√

β1ℓβ2u

.

By adding up J given by (19) and Fγ2 (γ) given by (6), Fγ3 (γ) is obtained asshown in (7).

Appendix B: Proof of Lemma 2

To derive an approximation for fγ3 (γ), we first obtain the first non-zero highorder derivative of fγ3 (γ). Using the definition of γ3, its PDF can be expressedas

fγ3 (γ) = fγ2 (γ) +

∫ ∞

0

∂[Fγ1 (ψ) fγ2 (γ + z)]

∂γdz (20)

where ψ = γ(γ+z)z . Hence, by applying the n-th order derivative of the product

in (20), i.e., Leibniz’s rule (see [10, eq. (0.42)]), the n-th order derivative of

52 Part I

fγ3 (γ) at zero value is given by

∂n

∂γnfγ3 (0) =

∂n

∂γnfγ2 (0) +

∫ ∞

0

n+1∑

i=0

(

n+ 1

i

)

× ∂iFγ1 (ψ)

∂γi

∣

∣

∣

∣

γ=0

∂n+1−ifγ2 (γ + z)

∂γn+1−i

∣

∣

∣

∣

γ=0

dz. (21)

From Lemma 1, it follows that fγ1 (γ) ≈ c1γd1 and fγ2 (γ) ≈ c2γ

d2 where

d1 = −1 +

κ1∑

ℓ=1

χ1ℓ, c1 =

∏κ1

ℓ=1 β1ℓ

d1!,

d2 = −1 +

κ2∑

ℓ=1

χ2ℓ, c2 =

∏κ2

ℓ=1 β2ℓ

d2!.

Applying the i-th order derivative of the composite function (see [10, eq.(0.430)]), results in

∂iFγ1 (ψ)

∂γi

∣

∣

∣

∣

γ=0

=

i∑

u=1

u−1∑

v=0

(

u

v

)

(−1)v

u!ψv(0)

∂uFγ1 (ψ)

∂ψu

∣

∣

∣

∣

γ=0

∂iψu−v

∂γu−v

∣

∣

∣

∣

γ=0

. (22)

Note that only the u = d1 +1 derivative makes∂uFγ1

(ψ)

∂ψu

∣

∣

∣

γ=0become non-zero

and equal to c1(d1)!. Further, observe that ψv(0) = [ (z+γ)γz ]v

∣

∣

∣

γ=0is non-zero

if and only if v = 0. Hence, we need to consider only u = dγ1 + 1 and v = 0for the compound summation in (22), yielding to

∂iFγ1 (ψ)

∂γi

∣

∣

∣

∣

γ=0

=c1

d1 + 1

∂iψd1+1

∂γd1+1

∣

∣

∣

∣

γ=0

. (23)

Using the result in (23), we can calculate the n-th derivative of fγ3 (γ) atγ = 0 by subdividing into the following three cases for d1 and d2:

• If d1 > d2: It is easy to see that ∂n

∂γn fγ2 (0) = c2(d2)! if n = d2 and∂n

∂γn fγ2 (0) = 0 if n < d2. For this particular case, we thus select n = d2.

Furthermore, since i ≤ (n + 1) = d2 + 1 < d1 + 1, it follows from (23)

that∂iFγ1

(ψ)

∂γi

∣

∣

∣

γ=0= 0, leading to the second summand of (21) to be

zero. Hence, ∂n

∂γn fγ3 (0) = c2(d2)! if n = d2 and zero if n < d2.


• If d1 = d2 = d: The same result as in the case of d1 > d2 holdsfor ∂n

∂γn fγ2 (0). However, the second summand of (21) has now a non-

zero value. The value of index i in (23) can be chosen as i = n +

1 = d + 1, yielding∂iFγ1

(ψ)

∂γi

∣

∣

∣

γ=0= c1d!. In addition, the expression

∂n+1−ifγ2(γ+z)

∂γn+1−i

∣

∣

∣

γ=0in (21) is now equal to fγ2 (z). Taking these results

into account, the second summand of the right-hand side of (21), i.e.,the integral, equals to c1d!. Then, it follows that ∂n

∂γn fγ3 (0) = (c1+c2)d!if n = d and zero if n < d.

• If d1 = d < d2: In this case, fγ3 (γ) can be rewritten as

fγ3 (γ) = fγ1 (γ) +

∫ ∞

0

∂[Fγ2 (ψ) fγ1 (γ + z)]

∂γdz. (24)

Using the same approach as in cases 1) and 2), yields ∂n

∂γn fγ3 (0) =

c1(d1)! and zero if n < d1.

Combining the three cases finalizes the proof.

Appendix C: Proof of Proposition 1

By applying a standard RVs transformation between γOSTBC and γ3, for whichγOSTBC = γ0

L1Rcγ3, and using Lemma 2, the PDF of γOSTBC can be approxi-

mated as

fγOSTBC(γ) ≈ c3

(

L1Rc

γ0

)d3+1

γd3 . (25)

Next, integrating (25) with respect to γ and setting γ = γthγ , yields (16).

The SEP can be approximated by utilizing the approach proposed in [17].In particular, since the up to d3-th order derivatives of fγOSTBC

(γ) are zero atγ = 0, the SEP for M -PSK modulation in the high SNR regime can be tightlyasymptotically approximated as

Perr ≈

d3+1∏

i=1

(2i− 1)

(d3 + 1)! (2g)d3+1

∂d3

∂γdγOSTBC

fγOSTBC(0) . (26)

By substituting (25) in (26) and after some algebraic manipulations, yields(17), completing thereby the proof.

54 Part I

References

[1] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time blockcodes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5,pp. 1456–1467, Jul. 1999.

[2] J. N. Laneman and G. W. Wornell, “Distributed space–time-coded pro-tocols for exploiting cooperative diversity in wireless networks,” IEEE

Trans. Inf. Theory, vol. 49, no. 10, pp. 2415–2425, Oct. 2003.

[3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversityin wireless networks: Efficient protocols and outage behavior,” IEEE

Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.


[5] Y. Song, H. Shin, and E.-K. Hong, “MIMO cooperative diversitywith scalar-gain amplify-and-forward relaying,” IEEE Trans. Commun.,vol. 57, no. 7, pp. 1932–1938, Jul. 2009.

[6] T. Q. Duong, H. Shin, and E.-K. Hong, “Effect of line-of-sight on dual-hop nonregenerative relay wireless communications,” in Proc. IEEE Veh.

Technol. Conf. (VTC) Fall, Baltimore, Maryland, Sep. 2007, pp. 571–575.

[7] P. Dharmawansa, M. R. McKay, and R. K. Mallik, “Dual hop MIMOrelaying with orthogonal space-time block codes,” in Proc. IEEE Int.

Conf. Commun. (ICC), Dresden, Germany, Jun. 2009, pp. 1–5.



[9] T. Q. Duong, H.-J. Zepernick, T. A. Tsiftsis, and V. N. Q. Bao, “Amplify-and-forward MIMO relaying with OSTBC over Nakagami-m fading chan-nels,” in Proc. IEEE Int. Conf. Commun. (ICC), Cape Town, SouthAfrica, May 2010, pp. 1–6.

[10] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Prod-

ucts, 6th ed. San Diego, CA: Academic, 2000.


[11] M. K. Simon and M.-S. Alouini, Digital Communication over Fading

Channels, 2nd ed. New York: Wiley, 2005.

[12] G. C. Alexandropoulos, N. C. Sagias, F. I. Lazarakis, and K. Berberidis,“New results for the multivariate Nakagami-m fading model with arbi-trary correlation matrix and applications,” IEEE Trans. Wireless Com-

mun., vol. 8, no. 1, pp. 245–255, Jan. 2009.

[13] G. K. Karagiannidis, N. C. Sagias, and T. A. Tsiftsis, “Closed-form statis-tics for the sum of squared Nakagami-m variates and its applications,”IEEE Trans. Commun., vol. 54, no. 8, pp. 1353–1359, Aug. 2006.

[14] M. R. McKay, A. J. Grant, and I. B. Collings, “Performance analysis ofMIMO-MRC in double-correlated Rayleigh environments,” IEEE Trans.


[15] A. Erdelyi, Higher Transcendental Functions. New York: McGraw-Hill,1953, vol. 1.

[16] Y. Li and S. Kishore, “Asymptotic analysis of amplify-and-forward relay-ing in Nakagami-fading environments,” IEEE Trans. Wireless Commun.,vol. 6, no. 12, pp. 4256–4262, Dec. 2007.

[17] A. Ribeiro, X. Cai, and G. B. Giannakis, “Symbol error probabilities forgeneral cooperative links,” IEEE Trans. Wireless Commun., vol. 4, no. 3,pp. 1264–1273, May 2005.

Part II

Part II

Keyhole Effect in MIMO AF Relay

Transmission with Space-Time Block Codes

Part II is published as

T. Q. Duong, H. A. Suraweera, T. A. Tsiftsis, H.-J. Zepernick, and A. Nal-lanathan “Keyhole Effect in MIMO AF Relay Transmission with Space-TimeBlock Codes,” IEEE Trans. Commun., Feb. 2012, under revision.

Based on

T. Q. Duong, H. A. Suraweera, T. A. Tsiftsis, H.-J. Zepernick, and A. Nal-lanathan “OSTBCS in MIMO AF Relay Systems with Keyhole and Corre-lation Effects,” in Proc. IEEE International Communications Conference,Kyoto, Japan, June 2011.

Keyhole Effect in MIMO AF Relay

Transmission with Space-Time Block Codes

Trung Q. Duong, Himal A. Suraweera, Theodoros A. Tsiftsis,

Hans-Jurgen Zepernick, and Arumugam Nallanathan

Abstract

In this paper, the effect of keyhole on the performance of multiple-input multiple-output (MIMO) amplify-and-forward (AF) relay net-works with orthogonal space-time block codes (OSTBCs) transmissionis investigated. In particular, we analyze the asymptotic symbol errorprobability (SEP) performance of a downlink communication systemwhere the amplifying processing at the relay can be implemented byeither the linear or squaring approach. Our tractable asymptotic SEPexpressions are shown to be very tight in the high signal-to-noise ra-tio regime which enable us to obtain both diversity and array gains.Our finding reveals that with condition nS > min(nR, nD), the linearapproach can provide the full achievable diversity gain of min(nR, nD)when only the second hop suffers from the keyhole effect, i.e., single key-hole effect (SKE), where nS, nR, and nD are the number of antennas atsource, relay, and destination, respectively. However, for the case thatboth the source-relay and relay-destination links experience the keyholeeffect, i.e., double keyhole effect (DKE), the achievable diversity orderis only one regardless of the number of antennas. In contrast, utiliz-ing the squaring approach, the overall diversity gain can be achieved asmin(nR, nD) for both SKE and DKE. An important observation corrob-orated by our studies is that for satisfying the tradeoff between perfor-mance and complexity, we should use the linear approach SKE and thesquaring approach for DKE.

1 Introduction

The multiple-input multiple-output (MIMO) technique has been consideredas a promising transmission scheme for future wireless systems as being in-

61

62 Part II

cluded in the extension of the 3GPP Long Term Evolution (LTE) standard.By deploying multiple antennas at transmitter and receiver ends, MIMO sys-tems exploit the rich scattering propagation environment of wireless links toenhance the performance [1]. In this ideal case of rich scattering, the chan-nel matrix is of full rank. As a consequence, a MIMO channel between annS-antenna source and an nD-antenna destination under such ideal conditionexhibits the maximum multiplexing gain of min(nS, nD) and diversity gain ofnSnD. In some realistic indoor and outdoor propagation conditions such ascorridors, crowded subways and tunnels, scattering objects can be randomlyarranged, which may cause a rank deficiency of the channel matrix. Thisdegeneration effect, so called keyhole or pinhole effect, significantly decreasesthe spatial multiplexing and diversity gains in MIMO systems [2, 3, 4]. As aresult, the keyhole effect reduces the MIMO channel capacity to that of single-input single-output (SISO) systems and the diversity gain is of min(nS, nD)order.

MIMO relay networks have attracted much interest because of their abil-ity to remarkably increase coverage and system performance [5, 6]. However,there has been limited research in the contemporary literature investigatingthe keyhole effect on the performance of MIMO relay networks. To the bestof the authors’ knowledge, only Souihli and Ohtsuki have very recently at-tempted to consider the effect of keyhole for MIMO relay systems [7, 8]. Inparticular, by considering a downlink cellular network in which the source-to-relay channel enjoys rich scattering while the source-to-destination andrelay-to-destination channels suffer from the keyhole phenomenon, they haveshown that using the relay can mitigate the effect of the keyhole in termsof channel capacity. However, [7, 8] have only focused on the multiplexinggain and the decode-and-forward (DF) relay protocol. In our previous work[9], we have investigated the cooperative diversity gain for a similar downlinksystem with orthogonal space-time block code (OSTBC) transmission. Wehave shown that the antenna correlation has no impact on the diversity gainof MIMO keyhole relay channels.

The OSTBC transmission over MIMO amplify-and-forward (AF) relaynetworks can be subdivided into two groups depending on the signal process-ing at the relay: 1) Linear Approach: the relay simply amplifies the source’sOSTBC matrix and forwards it to the destination without incurring any ad-ditional processing [10, 11] and 2) Squaring Approach: the relay performsthe squaring technique [12] to decompose the source’s OSTBC matrix intoindependent SISO signals and then re-encodes them by a new OSTBC matrixbefore forwarding to the destination [13, 14, 15]. It is important to note that

Keyhole Effect in MIMO AF Relay Transmission with Space-Time Block Codes 63

for the linear approach, the relay can operate in either semi-blind or channelstate information (CSI)-assisted modes. However, for the squaring approach,since the relay utilizes the CSI knowledge for decoupling the source’s OSTBCmatrix, it is reasonable to assume that the AF relay will operate using theCSI-assisted gain.

Several papers have investigated the performance of OSTBCs in AF relaysystems (see e.g., [10, 11, 13, 14, 15] and references therein). For the linearapproach, the exact symbol error probability (SEP) for semi-blind AF relayhas been derived for dual-hop fading channel in [10]. The final SEP expressionin [10] is not given in closed-form expression but as an integral whose integrandis expressed in terms of hypergeometric functions. In [11], several closed-formexpressions of the bit error rate (BER) for a specific number of antennas havebeen presented. For the squaring approach, the error rate performance hasbeen investigated for the cases when the destination has a single antenna[14] and multiple antennas [13]. The performance of OSBTC transmission inMIMO AF relay networks with best relay selection has been addressed in [15].

In this paper, we investigate the effect of MIMO keyhole channels on thecooperative diversity gain of relay networks. We consider the downlink co-operative communication where an nS-antenna base station S using OSTBCcommunicates with an nD-antenna mobile station D through the assistanceof an nR-antenna relay station R. To distinguish the advantage of each typeof signal processing at R, the linear approach is assumed to operate in semi-blind mode for low complexity processing while in the squaring approach theCSI-assisted AF relay is exploited to perform some additional complicatedprocessing. In addition, we consider both single keyhole effect (SKE) anddouble keyhole effect1 (DKE). For the case of SKE, the relay is assumed tobe a fixed base station (BS) and placed in some strategic location, leading thesource-to-relay link to enjoy rich scattering whereas the relay-to-destinationlink suffers poor scattering due to the mobility of the destination. For thecase of DKE, due to ad-hoc relay placement or in some specific networks suchas vehicle-to-vehicle communication [16, 17], we assume that both hops aresubject to the keyhole effect.

Our contributions include the following:

• We develop new analytical expressions to investigate the performance of

1In referring here to SKE, we imply that the first hop is assumed to be keyhole-free andsubject to Rayleigh fading. In contrast, for DKE both hops experience the keyhole effect.For a single-antenna relay system, the keyhole channel for mobile relay particularizes to adouble fading channel [16, 17].

64 Part II

AF systems with OSTBCs and keyhole effects for downlink communi-cation2. In particular, we investigate the system’s average SEP for bothlinear and squaring approaches.

• We present new asymptotic results which reveal the effect of keyholeon diversity and array gains of the considered system for both lin-ear and squaring approaches. For SKE, the relaying channel (source-relay-destination link) provides the full cooperative diversity gain ofmin(nR, nD) for both AF relaying approaches (linear and squaring).However, for the case of DKE, the linear approach cannot offer anydiversity gain, i.e., its diversity order is equal to that of a SISO channel.In contrast, for the squaring approach, the full achievable diversity gainremains as min(nR, nD).

• Our analysis gives additional insights for the system designer that forkeyhole relay channels, deployment of more than nD antennas at therelay, i.e., nR > nD, yields no diversity gain. More importantly, whenonly the second hop suffers from the keyhole effect, the relay shoulddeploy the linear approach to achieve the full diversity gain with lowcomplexity. When both hops experience the keyhole effect, the squaringapproach is more favorable than the linear approach because the lattercan only offer a unit diversity order.

Notation: A vector and a matrix are written as bold lowercase and uppercaseletters, respectively. The superscripts ∗ and † stand for complex conjugateand transpose conjugate, respectively. IIIn represents the n × n identity ma-trix and ‖AAA‖F defines Frobenius norm of the matrix AAA. E . and tr(·) is theexpectation operator and trace of a matrix, respectively. We use the nota-tion x ∼ CN (·, ·) to denote that x is complex circularly symmetric Gaussiandistributed. Let xxx ∈ Cm be the complex Gaussian distributed vector-variatewith mean vector vvv and variance matrix ΣΣΣ defined as xxx ∼ Nm (vvv,ΣΣΣ). Kν (·)is the modified Bessel function of the second kind [18, Eq. (8.432.3)]. Finally,let XXX ∈ Cm×n be the complex Gaussian distributed matrix-variate defined asXXX ∼ Nm,n (MMM,ΣΣΣ,ΦΦΦ) if vec(XXX†) is mn-variate complex Gaussian distributed

with mean vec(MMM †) and covariance ΣΣΣT ⊗ΦΦΦ, where ⊗ is the Kronecker productand vec(AAA) denotes the vector formed by stacking all the columns of AAA into acolumn vector.

2In this paper, we have assumed that nS > min(nR, nD) for all cases. Since the consideredsystem is downlink, i.e., nS > nD, the above condition is always satisfied. As a result,downlink communication implicitly indicates that nS > min(nR, nD).


S D

R

nS antennas

nR antennas

nD antennas

HSR HRD

Figure 1: System model for a dual-hop MIMO AF relay system consisting ofnS-antenna source, nR-antenna relay, and nD-antenna destination. (a) SKE:HHHRD is subject to the keyhole effect and HHHSR is keyhole-free. (b) DKE: BothHHHSR and HHHRD are subject to the keyhole effect.

2 System and Channel Models

We consider a communication network where a source terminal S commu-nicates with a destination D through the assistance of a relay station R asshown in Fig. 1. In this paper, we assume no direct link between the sourceand destination due to high pathloss and severe shadowing.

We next describe OSTBC transmission over a dual-hop AF relay net-work in detail. During an Lc-symbol interval, N log2M information bits aremapped to a sequence of symbols x1, x2, ..., xN selected from a M -ary phase-shift keying (M -PSK) or M -ary quadrature amplitude modulation (M -QAM)signal constellation S using Gray mapping with average transmit energy persymbol E

|xk|2

= P0. These symbols are then encoded with an OSTBC de-noted by an Lc ×nS transmission matrix GGG with code rate Rc whose elementsare linear combinations of x1, x2, ..., xN and their conjugate with the propertythat the columns of GGG are orthogonal [19].

We assume that the channel is subject to frequency-flat fading and isperfectly known at the destination but unknown at the source. The sourcewill transmit the OSTBC matrix XXX = GGGT to the relay in the first hop. Let usdenote Ps as the transmit power in nS antennas at the source so that the total

transmit power of an OSTBC block is E

‖XXX‖2F

= LcPs. The signal received

at the relay in the first hop is given by YYYSR = HHHSRXXX +WWWSR, where HHHSR ∼NnR,nS

(000,Ω1IIInR, IIInS

) is a random channel matrix, with Ω1 = E

|HHHSRij |2

66 Part II

for i = 1, 2, . . . , nR and j = 1, 2, . . . , nS, and WWWSR ∼ NnR,Nc(000, N0IIInR

, IIILc) is

an additive white Gaussian noise (AWGN) matrix, withN0 = E

|WWWSRij |2

.

2.1 Linear Approach

The received signal at the relay, YYYSR, is then multiplied by a fixed gainGLA, and retransmitted to the destination. The signal at the destinationin the second hop is given by YYYRD = HHHRDGLAYYYSR + WWWRD, where HHHRD ∼NnD,nR

(000,Ω2IIInD, IIInR

) is a random channel matrix, with Ω2 = E

|HHHRDkℓ|2

for k = 1, 2, . . . , nD and ℓ = 1, 2, . . . , nR, and WWWRD ∼ NnD,Lc(000, N0IIInD

, IIILc) is

an AWGN matrix, with N0 = E

|WWWRDkℓ|2

.Utilizing the orthogonal property of OSTBCs, the maximum likelihood

(ML) decoding metric can be decomposed into a sum of N terms, whereeach term depends on exactly one complex symbol xn, n = 1, 2, ..., N . Conse-quently, the detection of xn is decoupled from the detection of xp for n 6= pand the end-to-end instantaneous SNR can be written as [10]

γLA =γ

RcnS

tr

[

G2LAHHH

†RD

(

IIInD+G2

LAHHHRDHHH†RD

)−1

HHHRDHHHSRHHH†SR

]

, (1)

where γ = Ps/N0 is the average SNR and (1) is obtained from the fact

that E

‖XXX‖2F

= LcPs. Given the condition that the relay does not have

full channel knowledge of the first hop, GLA can be obtained as G2LA =

[nR (Ω1 + 1/γ)]−1 [10, 11]. When the relay is deployed with a single antenna,i.e., nR = 1, the channel matricesHHHSR andHHHRD for both hops become vectorshhhSR and hhhRD, respectively. The instantaneous SNR for the linear approachgiven in (1) can now be simplified as

γLA =γ

RcnS

‖hhhSR‖2F ‖hhhRD‖2

F

‖hhhRD‖2F + 1/G2

LA

. (2)

2.2 Squaring Approach

Using the squaring approach [12, 13, 14, 15] of OSTBC transmission, the

received matrix at the relay can be decomposed as yk = ‖HHHSR‖2F xk + ηk,

where k = 1, 2, . . . , N and ηk ∼ CN(

0, ‖HHHSR‖2FN0

)

. With the decomposed

symbols yk at hand, the relay generates an OSTBC matrix and then amplifiesit with an amplifying gain GSA before transmitting to the destination. For


the squaring approach, it is plausible to assume that the relay operates in theCSI-assisted mode and its amplifying gain is defined as GSA = (‖HHHSR‖F)−2

[13, 14, 15] by excluding the effect of the noise. The received signal at thedestination is given by YYYRD =HHHRDGSAXXX +WWWRD, where XXX is the re-encodedOSTBC at the relay after using the squaring approach3, whose element is thelinear combination of yk and y∗k for k = 1, 2, . . . , N . The end-to-end SNR isgiven by [13, 14, 15]

γSA =γ

RcnS

× ‖HHHSR‖2F ‖HHHRD‖2

F

‖HHHSR‖2F + ‖HHHRD‖2

F

. (3)

If nR = 1, it can be shown that the SNR expression given in (3) is equivalent to(2). This demonstrates that the two approaches exhibit the same performancewhen the system has a single-antenna relay. In the existing literature, tothe best of the authors’ knowledge, there is no comparison between the twoapproaches, even for the widely treated case of Rayleigh fading.

2.3 Keyhole Models and Statistics of ‖HHHRD‖2F and ‖HHHSR‖2

F

In this subsection, we first introduce the keyhole model adopted in this paperand some statistical properties of the two random variables (RVs) ‖HHHRD‖2

F

and ‖HHHSR‖2F which will be helpful for asymptotic SEP derivation.

2.3.1 Single Keyhole Effect

For this scenario, we assume that the relay terminal together with the sourceBS is part of a fixed infrastructure network, i.e., the relay has been installedat a strategic location by the network operator [7, 8, 9]. Hence, the S → R linkenjoys a rich-scattering environment, yielding the channel gain matrixHHHSR tobe of full rank. On the other hand, in downlink systems, D is considered asa mobile station (MS) and applicable to be in a poor scattering environment.In order to model this practical scenario of interest, we assume that the R →D link is a keyhole channel, i.e., HHHRD is of unit rank. With the keyholeassumption, the channel matrixHHHRD can be mathematically described as [21]

HHHRD = Ω2xxxRDyyy†RD. Here, xxxRD ∈ CnD and yyyRD ∈ CnR describe the scattering

environment at the destination and relay, respectively [21].

3In this paper, we apply the general OSTBC construction with the minimum delayand maximum achievable rate proposed in [20]. The code rates at S and R are given by⌈log2(nS)⌉+1

2⌈log2(nS)⌉ and⌈log2(nR)⌉+1

2⌈log2(nR)⌉ , respectively, where ⌈·⌉ denotes the ceiling function.

68 Part II

2.3.2 Double Keyhole Effect

In some unplanned situations, the deployment of the relay may be more or lessad hoc, e.g., the relay may be placed based on a rough knowledge of the cov-erage issues and traffic density (hotspots) in the network. In such cases, bothhops can be subject to poor scattering and thus we assume that bothHHHSR andHHHRD links of the network are under the influence of the keyhole effect. In thiscase, since the first-hop channel is now also under the keyhole effect, the cor-responding channel matrix therefore can be expressed as HHHSR =

√Ω1xxxSRyyy

†SR,

where xxxSR ∈ CnR , yyySR ∈ CnS are independent random vectors constitutingthe scattering environment at the relay and source, respectively [21]. It isimportant to note that for the case of the linear approach, the current formof γLA given in (1) is too complicated to be used for performance evaluationdue to the unit-rank property of HHHSR. Fortunately, we can utilize the unit-rank property to simplify the complex form given in (1), which facilitates ouranalysis. The derivation steps are shown in the sequel.

Since HHHSR =√

Ω1xxxSRyyy†SR, the SNR given in (1) can be rewritten as

γLA =γ

RcnS

G2LAΩ1 ‖yyySR‖2

F tr(

AAAxxxSRxxx†SR

)

, (4)

where AAA =HHH†RD

(

IIInD+G2

LAHHHRDHHH†RD

)−1

HHHRD. Owing to the fact that AAA is a

positive definite matrix, we can apply the eigen-decomposition yielding AAA =UUUΛΛΛUUU† where UUU is a unitary matrix and ΛΛΛ is the diagonal matrix containingthe eigenvalues of AAA. Since HHHRD =

√Ω2xxxRDyyy

†RD is a unit-rank matrix, AAA has

only one non-zero eigenvalue denoted as 1G2

LA+1/‖HHHRD‖2

F

leading (4) to become

γLA =γΩ1

RcnS

× |z|2 ‖yyySR‖2F ‖HHHRD‖2

F

‖HHHRD‖2F + 1/G2

LA

, (5)

where z is the first element of vector xxxSR, with xxxSR = UUU †xxxSR. Since UUU is a uni-tary matrix and xxxSR ∼ NnR

(000, IIInR), we can easily see that xxxSR ∼ NnR

(000, IIInR)

leads to z ∼ CN (0, 1). As compared to (1), the new expression of γLA is moretractable.

2.3.3 Statistics of ‖HHHRD‖2F and ‖HHHSR‖2

F

The statistics of ‖HHHRD‖2F and ‖HHHSR‖2

F applicable under keyhole conditionswill be utilized in our subsequent performance analysis. Due to symmetry


between ‖HHHRD‖2F and ‖HHHSR‖2

F, let us denote AB ∈ SR,RD and ℓ ∈ 1, 2 for

the sake of brevity. SinceHHHAB =√

ΩℓxxxAByyy†AB, we then can express ‖HHHAB‖2

F =

Ωℓ ‖xxxAB‖2F ‖yyyAB‖2

F. As xxxAB and yyyAB are statistically independent, the PDF of

‖HHHAB‖2F when subject to the keyhole effect is given by [4]

f‖HHHAB‖2F

(z) =2z(nA+nB)/2−1

Γ(nA)Γ(nB)Ω(nA+nB)/2ℓ

KnB−nA

(

2

√

z

Ωℓ

)

. (6)

Since ‖HHHAB‖2F is the product of two RVs, the CDF of ‖HHHAB‖2

F can be written

as F‖HHHAB‖2F

(z) =∫ ∞

0f‖xxxAB‖2

F(w)F‖yyyAB‖2

F

(

zΩℓw

)

dw. Note that the CDF of

‖yyyAB‖2F can be given in the form of elementary functions by applying [18,

Eq. (8.352.4)]. Then, utilizing the result of [18, Eq. (3.471.9)], we get the

CDF of ‖HHHAB‖2F as

F‖HHHAB‖2F

(z) = 1 −nA−1∑

k=1

2

Γ(nB)k!

(

z

Ωℓ

)(nB+k)/2

KnB−k

(

2

√

z

Ωℓ

)

. (7)

3 High-SNR SEP Analysis

It is noted that for the SKE and DKE cases considered, an exact closed-formSEP analysis is not possible. Moreover, using our subsequent analysis, theexact SEP can be evaluated using numerical integration. In order to obtainkey insights and how different parameters affect the system parameters, wetherefore investigate the high SNR SEP behavior that yields the array anddiversity gains of MIMO AF systems with the keyhole effect.

3.1 Single Keyhole Effect with Linear Approach

For the dual-hop MIMO AF relaying with linear approach, from (1), the mo-ment generating function (MGF) can be obtained from the definition MLA (s) =EγLA

e−sγLA as [10, 11]

MLA (s) = EHHHRD

det(

IIInD+G2

LAHHHRDHHH†RD

)

det(

IIInD+G2

LA

(

1 + sγΩ1

RcnS

)

HHHRDHHH†RD

)

nS

. (8)

70 Part II

Let us denote x as a non-zero eigenvalue of HHHRDHHH†RD. Since HHHRD is of unit

rank, it is easy to see that there exists only one non-zero eigenvalue of x =‖HHHRD‖2

F. Using this property and from (6), we can rewrite (8) as

MγLA(s) =

∫ ∞

0

2(Ω2γ)−(nR+nD)/2

Γ(nR)Γ(nD)

[

1 +G2

LAΩ1st

RcnS (1 +G2LAγ

−1t)

]−nS

× t(nR+nD)/2−1KnD−nR

(

2

√

t

γΩ2

)

dt. (9)

To the best of the authors’ knowledge, (9) has no close-form solution. As aresult, we present the asymptotic SEP for single keyhole effect with linearapproach in the following Theorem.

Theorem 1 In the high SNR regime, the SEP expression for MIMO AF re-

laying with linear approach and single keyhole effect can be approximated as

P (SRD)e

(large γ)≈ c1

(

Ω1Ω2G2LAγ

RcnS

)−min(nR,nD)

× Γ(nS − min(nR, nD))Γ(min(nR, nD))

Γ(nS)Γ(nR)Γ(nD)(10)

with the condition that nS > min(nR, nD) and c1 is a constant depending on

the modulation scheme. For example, for M -PSK modulation, c1 is defined

as

c1 =

1π

∫ π− πM

0

(

g

sin2 θ

)−nR for nR = nD,

×[

ln(

Ω1Ω2G2LAγg

RcnS sin2 θ

)

− ψ(nR) + ψ(nS − nR)]

dθ

Γ(|nD−nR|)π

∫ π− πM

0

(

gsin2 θ

)−min(nR,nD)dθ for nR 6= nD.

(11)

Proof See Appendix A.

Since we consider downlink communication, it is reasonable to assume thatnS > nD. As can be observed from (10), the SEP is inversely proportional toγmin(nR,nD)4. Hence, the full achievable diversity gain provided by the fixedrelay link using linear approach is min(nR, nD). In addition, as can be observedfrom (10), the array gain can be clearly obtained.

4Although the asymptotic SEP is given in the form of the product of polynomial andlogarithm functions with respect to the average SNR γ [22, 23], the diversity order is solely

determined by the polynomial term since log log γ

log γcan be neglected as γ → ∞.


3.2 Double Keyhole Effect with Linear Approach

In the case of the double keyhole channel, as can be seen from (5), the exactPDF derivation of γLA requires the statistics of multiple products and divi-sion of RVs. In the considered problem, these expressions are not trivial toobtain and do not lend mathematical tractability for further manipulation.Therefore, we will utilize the upper and lower bounds of γLA for the analysisinstead. Using (5), the instantaneous end-to-end SNR when both hops un-

dergo keyhole can be rewritten as γLA =(

1γ1

+ c2

γ1γ2

)−1

, where c2 = γ/G2LA,

γ1 = Ω1γRcnS

|z|2 ‖yyySR‖2F, and γ2 = γ ‖HHHRD‖2

F . Hence, it is convenient for us tointroduce the lower and upper bounds of γLA as

γ(lo)LA ,

1

2min(γ1,

γ1γ2

c2) ≤ γLA ≤ min(γ1,

γ1γ2

c2) , γ

(up)LA . (12)

Theorem 2 In the high SNR regime, the SEP expression for MIMO AF re-

laying with linear approach and double keyhole effect can be lower and upper

bounded as

c3γ

≤P (SRD)e ≤ 2c3

γ, (13)

where

c3 =

[

ǫ

Ω2G2LA(nR − 1)(nD − 1)

+ ǫ

] RcnS

Ω1(nS − 1)πg

[

(M − 1)π

2M+

sin(2π/M)

4M

]

.

(14)


Based on (13), we can observe that the slope of the SEP curve in the high SNRregime is one. In other words, when both hops experience the keyhole effect,a relay using the linear approach exhibits an unit diversity order regardless ofthe number of antennas.

3.3 Single Keyhole Effect with Squaring Approach

With the squaring approach, the instantaneous SNR incurred by the dual-hopchannel can be given in the form of the harmonic mean of two independentRVs as in (3) and rewritten as follows:

γSA =1

RcnS

γ2γ3

γ2 + γ3, (15)

72 Part II

where γ3 = γ ‖HHHSR‖2F. In this case, since the first hop, i.e., S → R, is keyhole-

free, the squared Frobenius norm of HHHSR, ‖HHHSR‖2F, is the sum of nSnR inde-

pendent and identically distributed exponential RVs. Then, the PDF of γ3

can be approximated as [24]

fγ3 (γ3)(large γ)

≈ γnSnR−13

Γ(nSnR)(γ)nSnR−1. (16)

Depending on the relationship between nR and nD, the asymptotic SEP ex-pression for MIMO AF relaying with squaring approach and single keyholeeffect can be presented in the following Theorem.

Theorem 3 When nR 6= nD, in the high SNR regime, the SEP is approxi-

mated as

P (SRD)e

(large γ)≈

∏min(nR,nD)i=1 (2i− 1)

min(nR, nD)

Γ(|nD − nR|)Γ(nR)Γ(nD)

( RcnS

2gΩ2γ

)min(nR,nD)

. (17)

When nR = nD, in the high SNR regime, the SEP can be upper and lower

bounded as

c4(1)

γnR≤ P (SRD)

e ≤ c4(2)

γnR, (18)

where c4(⋆) is a constant expressed as

c4(⋆) =1

πΓ(nR)

(RcnS

Ω2g

)nR∫ π−π/M

0

(

⋆ sin2 θ)nR

×[

ln

(

Ω2γg

⋆RcnS sin2 θ

)

− ψ(nR)

]

dθ. (19)

Proof See Appendix C.

By combining (17) and (18), we see that when the fixed relay terminal usesthe squaring approach, the full achievable diversity order is min(nR, nD).

3.4 Double Keyhole Effect with Squaring Approach

In this case where both hops are subject to the keyhole effect, the PDF andCDF of γ3 can be obtained from (6) and (7), respectively, by applying a Jaco-bian transformation. Since we consider a downlink communication scenario,


i.e., nS > nR, from (25) fγ3 (γ) can be approximated as

fγ3 (γ)(large γ)

≈ Γ(nS − nR)

Γ(nS)Γ(nR)

γnR−1

(Ω1γ)nR. (20)

Again, similarly as in Sections 3.1, 3.2, and 3.3, by distinguishing separatecases due to the approximation of the modified Bessel function of the secondkind, the asymptotic SEP expression for MIMO AF relaying with squaringapproach and double keyhole effect can be presented in the following Theorem.

Theorem 4 In the high SNR regime, the SEP expression can be approximated

for the cases nR > nD and nR < nD, respectively, as follows:

P (SRD)e

(large γ)≈

∏nD

i=1(2i− 1)Γ(nR − nD)

Γ(nR)Γ(nD + 1)

( RcnS

2gΩ2γ

)nD

. (21)

P (SRD)e

(large γ)≈

∏nR

i=1(2i− 1)

Γ(nR + 1)

[

Γ(nD − nR)

Γ(nD)ΩnR

2

+Γ(nS − nR)

Γ(nS)ΩnR

1

] (RcnS

2gγ

)nR

. (22)

When nR = nD, in the high SNR regime, the SEP expression can be lower and

upper bounded as

c5(1)

γnR≤ P (SRD)

e ≤ c5(2)

γnR, (23)

where c5(⋆) is a constant given by

c5(⋆) =(RcnS)

nR

πΓ(nR)

∫ π−π/M

0

(

⋆ sin2 θ

g

)nR

×

ln(

Ω1γg

⋆RcnS sin2 θ

)

− ψ(nR)

ΩnR

2

+Γ(nS − nR)

ΩnR

1

dθ. (24)

Proof See Appendix D.

By combining (21), (22), and (23), we can observe that the diversity order ismin(nR, nD). Therefore, compared with the unit diversity order achieved withthe linear approach (cf. Section 3.2), a larger diversity order can be achievedwith the squaring approach.

74 Part II

0 5 10 15 20 25 30 35 40

10-8

10-6

10-4

10-2

100

Exact (4,2,2) Asymptotic approximation (4,2,2) Exact (5,4,3) Asymptotic approximation (5,4,3)

8-PSK

SNR (dB)

Sym

bol E

rror

Pro

babi

lity

Figure 2: SEP of (4, 2, 2) and (5, 4, 3)-MIMO AF relay systems for 8-PSKmodulation versus SNR when only the second hop experiences the keyholeeffect and the relay uses the linear approach.


In this section, we investigate the keyhole effect on the SEP performance ofOSTBC MIMO AF relaying with the help of the analytical derivation devel-oped in Section 3. To illustrate the tightness of the asymptotic result, we alsoprovide exact numerical results, which are obtained from numerical integra-tion. For example, in the case of fixed relay with the linear approach, from(9), we can evaluate the exact SEP using numerical integration techniques.However, these SEP expressions are in the form of double or triple integralsand do not reveal any insight into the system’s performance. Moreover, sim-ulation results also confirm our analytical results but are not shown here toavoid clutter in the figures.

In all examples, results are shown for 8-PSK modulation and unit vari-ances (Ω1 = Ω2 = 1). We consider different numbers of antennas at S, R, D

denoted as (nS, nR, nD), where nS > min(nR, nD). In addition, the OSTBCsare generated from [20], e.g., the code-rates for the case of two and three


0 5 10 15 20 25 30 35 4010-4

10-3

10-2

10-1

100

Exact (4,2,2) Upper bound (4,2,2) Exact (5,4,3) Upper bound (5,4,3)

8-PSK

SNR (dB)

Sym

bol E

rror

Pro

babi

lity

Figure 3: SEP of (4, 2, 2) and (5, 4, 3)-MIMO AF relay systems for 8-PSKmodulation versus SNR when both hops experience keyhole effect and therelay uses the linear approach.

transmit antennas are one and 3/4, respectively.For the linear approach, the SEP when only the second hop experiences the

keyhole effect is shown in Fig. 2, where we compare asymptotic approximationwith exact numerical results for two different antenna configurations, e.g.,(4, 2, 2) and (5, 4, 3). As expected, the asymptotic curves tightly converge tothe exact curves in the high SNR regime. We can see that the slopes of SEPcurves are regulated by the minimum number of antennas at the relay anddestination. Fig. 3 displays the SEP performance with DKE and the squaringapproach. We have plotted the upper asymptotic bounds for the two differentMIMO AF relay systems as in the above example. Clearly, as observed fromthe plots, the upper bound becomes very tight in the high SNR regime andcan be considered as the asymptotic approximation. For further comparison,the two systems exhibit the same diversity gain as expected. We see thatas the number of antennas increases, i.e., from (4, 2, 2) to (5, 4, 3), the slopesof SEP curves remain unchanged and equal to one, which agrees with ouranalytical conclusion in Section 3.3.

For the squaring approach, Fig. 4 shows the SEP performance of the two

76 Part II

0 5 10 15 20 25 30 35 4010-10

10-8

10-6

10-4

10-2

100

Exact (4,3,2) Asymptotic approximation (4,3,2) Exact (5,4,4) Asymptotic approximation (5,4,4)

8-PSK

SNR (dB)

Sym

bol E

rror

Pro

babi

lity

Figure 4: SEP of (4, 3, 2) and (5, 4, 4)-MIMO AF relay systems for 8-PSKmodulation versus SNR when only the second hop experiences the keyholeeffect and the relay uses the squaring approach.

different MIMO AF relay systems, i.e., (4, 3, 2) and (5, 4, 4), when only thesecond hop experiences the keyhole effect. The SEP performance improves asthe min(nR, nD) increases. Again, we observe an excellent agreement betweenthe asymptotic and exact curves in the high SNR regime. In Fig. 5, weplot the average SEP when both hops are subject to keyhole fading. Threespecific system configurations are used, i.e., (5, 4, 4), (5, 4, 3), and (4, 2, 3),corresponding to the three conditions in the Section 3.4 (i.e., nR = nD, nR >nD, and nR < nD). In the three cases, our asymptotic results are seen toconverge to the respective exact curves for relatively high SNR levels (SNR> 20 dB). As expected, the best performance among the three examples isachieved for (5, 4, 4).

To further demonstrate the cooperative diversity derived in the previoussection, we plot the SEP for (6, nR, 2) with the linear approach and varying nR

from three to five and for (7, nR, 3) with the squaring approach and varyingnR from four to six. For comparison, we also plot the SEP for single-hop non-cooperative communications with keyhole and the SEP for single antenna


0 5 10 15 20 25 30 35 4010-10

10-8

10-6

10-4

10-2

100

Exact (5,4,4) Asymptotic approximation (5,4,4) Exact (5,4,3) Asymptotic approximation (5,4,3) Exact (4,2,3) Asymptotic approximation (4,2,3)

8-PSK

SNR (dB)

Sym

bol E

rror

Pro

babi

lity

Figure 5: SEP of (4, 2, 3), (5, 4, 3), and (5, 4, 4)-MIMO AF relay systems for8-PSK modulation versus SNR when both hops experience the keyhole effectand the relay uses the squaring approach.

relay. Here, we assume that R is located half-way between S and D, whichresults in the channel mean power for the direct link as Ω0 = 1/16. As canbe observed from Fig. 6, increasing the number of antennas at relays whilekeeping nD unchanged results in no diversity enhancement. This observationis in line with our analytical result derived in previous section as the fullachievable diversity gain is min(nR, nD). In fact, under such a severe conditionit is unnecessary to deploy a large number of antennas at the relay, onlyrequiring nR = nD. More importantly, we can see that direct communication,i.e., (nS, nD) = (7, 3), exhibits the same diversity gain as MIMO AF relayingwith the squaring approach, i.e., (7, nR, 3). However, relaying link with thesquaring approach significantly outperforms the direct link in terms of arraygain. Also, when SNR ≤ 35 dB the SEP of direct communication is inferiorto that of MIMO AF relay with the linear approach, which clearly highlightsthe advantage of using relays in keyhole fading.

78 Part II

0 5 10 15 20 25 30 35 4010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

nR=4,5,6

Linear approach (6,nR,2)

Squaring approach (7,nR,3)

Keyhole-direct (nS=7,n

D=3)

nS=n

R=n

D=1 Rayleigh

8-PSK

1=

2=1

0=1/16

R is half-way between S and D

nR=3,4,5

SNR (dB)

Sym

bol E

rror

Pro

babi

lity

Figure 6: SEP of (6, nR, 2) and (7, nR, 3)-MIMO AF relay systems for 8-PSKmodulation versus SNR when only the second hop experiences the keyholeeffect. For comparison, SEPs for direct link with keyhole and single antennarelaying link are also shown.

5 Conclusion

We have investigated the effect of keyhole on the performance of downlink AFrelay systems with OSTBC transmission. In particular, we have investigatedthe effect of keyhole on the SEP performance when the AF relay operates ineither linear or squaring approaches. Our tractable asymptotic SEP expres-sions, shown to be very tight with the exact results in the high SNR regime,reveal both diversity and array gains and provide several important insightsfor radio system designers. For the case when only the second hop suffersfrom the keyhole effect, we have suggested to apply the linear approach atthe relay for obtaining the full achievable diversity gain of min(nR, nD) whilekeeping low complexity for the relay. In contrast, for a more severe scenariowhere both hops are keyhole channels, the squaring approach should be usedto keep the diversity order as min(nR, nD) since the linear approach does notoffer any diversity gain.



To study the high SNR performance, we make use of the fact that for smallvalues of z, Kn (z) can be approximated as [25]

Kn (z)(small z)

≈

ln(

2z

)

for n = 0,Γ(|n|)

2

(

2z

)|n|for n 6= 0.

(25)

By substituting (25) into (9), we obtain a tight approximation for MγLA(s)

in the high SNR regime. Note that depending on the value of nR and nD,KnD−nR

(·) can be differently approximated as in (25). Therefore, we willseparately investigate two cases as follows: For the case nR = nD, replacing

KnD−nR

(

2√

tγΩ2

)

by 12 ln

(

γΩ2

t

)

and neglecting the small term relative to γ

in the integrand of (9) results in

MγLA(s)

(large γ)≈ (Ω2γ)−nR

Γ(nR)Γ(nD)

[∫ ∞

0

ln(Ω2γ)tnR−1

(

1 +G2

LAΩ1s

RcnSt)nS

dt

−∫ ∞

0

tnR−1 ln(t)(

1 +G2

LAΩ1s

RcnSt)nS

dt

]

. (26)

The first and second integral in the above equation can be easily computed byapplying the results of [18, Eq. (3.194.3)] and [26, Eq. (2.6.4.7)], respectively,with the condition that nS > nR. After several algebraic manipulations, theMGF of γLA can be written as

MγLA(s)

(large γ)≈

(

Ω1Ω2G2LAγs

RcnS

)−nR Γ(nS − nR)

Γ(nS)Γ(nR)

×[

ln

(

Ω1Ω2G2LAγs

RcnS

)

− ψ(nR) + ψ(nS − nR)

]

. (27)

For the case nR 6= nD, we can find the MGF of γLA as

MγLA(s)

(large γ)≈ (Ω2γ)

−min(nR,nD)

Γ(nR)Γ(nD)

∫ ∞

0

tmin(nR,nD)−1

(

1 +G2

LAΩ1s

RcnS

t

)−nS

dt.

(28)

80 Part II

Note that the integral (28) converges when nS > min(nR, nD) which thenyields

MγLA(s)

(large γ)≈

(

Ω1Ω2G2LAγs

RcnS

)−min(nR,nD)

× Γ(nS − min(nR, nD))Γ(min(nR, nD))Γ(|nD − nR|)Γ(nS)Γ(nR)Γ(nD)

. (29)

By combining the two cases, the asymptotic SEP of a dual-hop channel, i.e.,S → R → D link, when the fixed relay terminal operates in linear mode canbe expressed as (10), which completes our proof.

Appendix B: Proof of Theorem 2

We first derive the upper bound and note that the derivation of lower bound

can be followed accordingly. By definition, the CDF of γ(up)LA is given by

Fγ(up)LA

(γ) = Pr

(

γ1 < γ, γ1 <γ1γ2

c2

)

+ Pr

(

γ1γ2

c2< γ,

γ1γ2

c2< γ1

)

︸︷︷︸

L

. (30)

In addition, we have

L = Pr (γ1 < γ, γ2 < c2) + Pr (γ1 > γ, γ1γ2 < γc2) . (31)

As a result, the CDF of γ(up)LA can be rewritten as

Fγ(up)LA

(γ) = Pr (γ1 < γ, γ2 > c2) + Pr (γ1 < γ, γ2 < c2)︸︷︷︸

=Pr(γ1<γ)

+ Pr (γ1 > γ, γ1γ2 < γc2) , (32)

which then yields

Fγ(up)LA

(γ) = Fγ1 (γ) + Pr

(

γ < γ1 <γc2γ2

)

. (33)

Since further manipulations of the exact Fγ(up)LA

(γ) is cumbersome, we apply

the following approximation

Fγ(up)LA

(γ) ≈ Pr (γ2 < c2) Pr

(

γ1γ2

c2< γ

)

+ Pr (γ2 ≥ c2) Pr (γ1 < γ) . (34)


Since γ1 is the product of exponent and chi-square RVs, its PDF and CDF isgiven by, respectively

fγ1 (x) =2x(nS−1)/2

Γ(nS)[Ω1γ/(RcnS)](nS+1)/2KnS−1

(

2

√

x

Ω1γ/(RcnS)

)

, (35)

Fγ1 (x) = 1 − 2xnS/2

Γ(nS)[Ω1γ/(RcnS)]nS/2KnS

(

2

√

x

Ω1γ/(RcnS)

)

. (36)

Moreover, the statistics of γ2 can be obtained immediately from the PDF andCDF of ‖HHHRD‖2

F given in (6) and (7), respectively. Our aim now is to calculate

the MGF of γ(up)LA . To do so, we differentiate (34) with respect to γ to obtain

the PDF of γ(up)LA as

fγ(up)LA

(γ|γ2) =ǫc2γ2fγ1

(

c2γ

γ2|γ2

)

+ ǫfγ1 (γ) , (37)

where ǫ = Fγ2 (c2) can be deduced from (7) and ǫ = 1 − ǫ. As the MGF of

γ(up)LA is the Laplace transform of its PDF, we have

Mγ(up)LA

(s) = ǫ

∫ ∞

0

Mγ1

(

sγ2

c2

)

fγ2 (γ2) dγ2 + ǫMγ1 (s) , (38)

where the first summand in (38) is obtained by exchanging the order of doubleintegral, which is originated from the marginal PDF, with respect to the twovariables γ and γ2; and Mγ1 (s) is the MGF of γ1 shown as follows:

Mγ1 (s) =1

Ω1γs/(RcnS)Ψ

(

1, 2 − nS,1

Ω1γs/(RcnS)

)

, (39)

where (39) follows from (35) and Ψ (a, b; z) is the confluent hypergeometricfunction [18, Eq. (9.211.4)]. Since nS ≥ 2, from (39) and using the fact that

Ψ (a, b; z)(small z)

≈ Γ(1−b)Γ(1+a−b) if b ≤ 0 [27], we get

Mγ1 (s)(large γ)

≈ 1

(nS − 1)Ω1γs/(RcnS). (40)

82 Part II

By substituting (40) and (6) into (38), the integral can be approximated as

∫ ∞

0

Mγ1

(

sγ2

c2

)

fγ2 (γ2) dγ2

(large γ)≈ 2RcnS

Ω1G2LA(Ω2γ)(nR+nD)/2Γ(nS)Γ(nR)Γ(nD)s

×∫ ∞

0

γ(nR+nD)/2−22 KnD−nR

(

2

√

γ2

Ω2γ

)

dγ2

=RcnS

Ω1Ω2G2LA(nS − 1)(nR − 1)(nD − 1)γs

, (41)

where (41) is obtained by making use of [18, Eq. (6.561.16)]. Next, combining

(40) and (41) with (38), the high SNR approximation of the MGF of γ(up)LA

can be rewritten as

Mγ(up)LA

(s) =

[

ǫ

Ω2G2LA(nR − 1)(nD − 1)

+ ǫ

] RcnS

Ω1(nS − 1)γs. (42)

The asymptotic MGF of γ(lo)LA can be easily obtained from (42) by utilizing the

fact that Mγ(lo)LA

(s) = Mγ(up)LA

(

12s

)

. Hence, the asymptotic SEP can be shown

in (13), which finalizes the proof.

Appendix C: Proof of Theorem 3

We first consider for the case nR 6= nD. From (25) and (6), the PDF of γ2 canbe approximated as

fγ2 (γ2)(large γ)

≈ Γ(|nD − nR|)Γ(nR)Γ(nD)

γmin(nR,nD)−12

(Ω2γ)min(nR,nD). (43)

Following the same approach as in [24], the SEP can be approximated by thefirst non-zero high order derivative of fZ (z) as

P (SRD)e

(large γ)≈

∏d+1i=1 (2i− 1)

(d+ 1)!

(RcnS

2g

)d+1∂d

∂zdfZ (0) , (44)

where Z = γ2γ3

γ2+γ3. Our main objective now is to study the behavior of fZ (z)

in the high SNR regime. Let us write the PDF of Z as

fZ (z) = fγ2 (z) +

∫ ∞

0

∂[

Fγ3

(

z2

x + z)

fγ2 (z + x)]

∂zdx. (45)


Applying the Leibniz rule, the d-th order derivative of fZ (z) at zero valuecan be expressed as [18, Eq. 0.42]

∂d

∂zdfZ (0) =

∂d

∂znfγ2 (0)

︸︷︷︸

I1

+

∫ ∞

0

d+1∑

i=0

(

d+ 1

i

) ∂iFγ3

(

z2

x + z)

∂zi

∣

∣

∣

∣

∣

∣

z=0

∂d+1−ifγ2 (z + x)

∂zd+1−i

∣

∣

∣

∣

z=0

dx

︸︷︷︸

I2

. (46)

The first summand I1 can be easily obtained from (43) as

I1 =

Γ(|nD−nR|)Γ(nR)Γ(nD)

Γ(min(nR,nD))

(Ω2γ)min(nR,nD) if d = min(nR, nD) − 1,

0 if d < min(nR, nD) − 1.(47)

To calculate the second summand I2, we use the i-th order derivative of thecomposite function [18, Eq. 0.430] which results in

∂iFγ3

(

t = z2

x + z)

∂zi

∣

∣

∣

∣

∣

∣

z=0

=

i∑

u=1

u−1∑

v=0

(

u

v

)

(−1)v

u!tv(0)

∂uFγ3 (t)

∂tu

∣

∣

∣

∣

z=0

∂itu−v

∂zu−v

∣

∣

∣

∣

z=0

.

(48)

It is observed that only v = 0 makes tv(0) = ( z2

x + z)v∣

∣

∣

γ=0being non-zero.

From (16), the highest order of Fγ3 (t) is nSnR which allows us to select u =nSnR. Hence, (48) becomes

∂iFγ3

(

t = z2

x + z)

∂zi

∣

∣

∣

∣

∣

∣

z=0

=1

Γ(nSnR + 1)(γ)nSnR−1

∂itnSnR

∂znSnR

∣

∣

∣

∣

z=0

. (49)

It is obvious that I2 equals to zero when d ≤ min(nR, nD)−1. This is becausewith i ≤ d + 1 ≤ min(nR, nD) < nSnR, (49) equals to zero. Hence, we willselect d = min(nR, nD)−1. By jointly considering the aforementioned results,we can obtain the asymptotic SEP as in (17).

Next we consider for the case nR = nD by introducing the lower and upperbound of γSA as

γ(lo)SA ,

1

2RcnS

min(γ2, γ3) ≤ γSA ≤ 1

RcnS

min(γ2, γ3) , γ(up)SA . (50)

84 Part II

As observed from (50), the bounds in this case involve two independent RVs asopposed to (12) which is related to two dependent RVs. Hence, it is importantto note that although we use the lower and upper bounds as in Section 3.2,the mathematical derivations will proceed differently. We are interested in

deriving the MGF of γ(up)SA since the MGF of γ

(lo)SA can be followed accordingly.

Let us denote T = min(γ2, γ3). It is easy to see that as

Mγ(up)SA

(s) = Mγ2

(

s

RcnS

)

+ Mγ3

(

s

RcnS

)

−∫ ∞

0

fγ2 (t)Fγ3 (z) e− st

RcnS dt

−∫ ∞

0

fγ3 (t)Fγ2 (t) e− st

RcnS dt. (51)

To proceed, we substitute (7) into (51) and make use of the fact that γ3 is thesum of nSnR independent exponential variates. After several manipulations,

the MGF of γ(up)SA can be re-expressed as

Mγ(up)SA

(s) =

∫ ∞

0

Afγ2 (t) e− st

RcnS dt

︸︷︷︸

I3

+

∫ ∞

0

Bfγ3 (t) e− st

RcnS dt

︸︷︷︸

I4

, (52)

where A and B are, respectively, given by

A =

nSnR−1∑

k=0

(

t

Ω1γ

)ke−

tΩ1γ

k!,

B =

nR−1∑

k=0

2

Γ(nD)k!

(

t

Ω2γ

)(nD+k)/2

KnD−k

(

2

√

t

Ω2γ

)

. (53)

When nR = nD, using (25) and (6), an approximate expression for the PDFof γ2 can be obtained as

fγ2 (t)(large γ)

≈tnR−1 ln

(

Ω2γt

)

Γ(nR)2(Ω2γ)nR. (54)

Substituting (53) and (54) into (52) yields

I3

(large γ)≈

nSnR−1∑

k=0

1

Γ(nR)2Ωk1ΩnR

2 k!γnR+k

∫ ∞

0

tnR+k−1e−( 1

Ω1 γ+ s

RcnS)t

ln

(

Ω2γ

t

)

dt,

(large γ)≈

nSnR−1∑

k=0

Γ(nR + k)[

ln(

Ω2γsRcnS

)

− ψ(nR + k)]

Γ(nR)2Ωk1ΩnR

2 k!

(RcnS

γs

)nR+k

, (55)


where (55) is obtained by using 1Ω1γ

(large γ)≪ 1

RcnSand [18, Eq. (4.352.1)].

Similarly, we get the following approximation for I4:

I4

(large γ)≈

nR−1∑

k=0

Γ(nR − k)Γ(nSnR + k)

Γ(nR)Γ(nSnR)Ωk2ΩnSnR

1 k!

(RcnS

γs

)nSnR+k

. (56)

Since I3 and I4 are proportional to(

1γ

)nR+k

and(

1γ

)nSnR+k

with integer k,

respectively, I4 can be neglected as compared to I3. Then by selecting theindex k = 0 (higher values of k can be omitted) for I3, the approximated

MGF of γ(up)SA is given by

Mγ(up)SA

(s)(large γ)

≈ln

(

Ω2γsRcnS

)

− ψ(nR)

Γ(nR)ΩnR

2

(RcnS

γs

)nR

. (57)

Using (57), the SEP behavior in the high SNR regime pertaining to the caseof nR = nD can be given by (18), which concludes our proof.

Appendix D: Proof of Theorem 4

When nR > nD, by following the same approach as in Section 3.3, we caneasily obtain (21). For the case nR < nD, we can obtain I1 given in (46) as

I1 =Γ(nD − nR)

Γ(nD)(Ω2γ)nRif d = nR − 1, and I1 = 0 if d < nR − 1. (58)

In Section 3.3, I2 given in (46) is zero. In contrast, I2 now becomes non-zeroand the index i can be selected as i = d + 1 = nR, which then allows us

to rewrite the right-hand-side of (48) as Γ(nS−nR)Γ(nS)(Ω1γ)nR

. In addition, the term

∂d+1−ifγ2 (z+x)

∂zd+1−i

∣

∣

∣

z=0of I2 given in (46) now becomes fγ2 (x) and is given by

I2 = Γ(nS−nR)Γ(nS)(Ω1γ)nR

. Utilizing this result and pulling (58) together with (46)

and (44), the asymptotic SEP is expressed as (22). The derivation for thecase nR = nD follows the same pattern as in Section 3.3 and therefore isomitted for brevity. Specifically, the result of I3 can be shown as

I3

(large γ)≈

[

ln(

Ω1γsRcnS

)

− ψ(nR)]

Γ(nR)ΩnR

2

(RcnS

γs

)nR

. (59)

86 Part II

In contrast to the derivation in Section 3.3, I4 can not be neglected. In fact,it is comparable with I3 and can be expressed as

I4

(large γ)≈ Γ(nS − nR)

Γ(nR)ΩnR

1

(RcnS

γs

)nR

. (60)

which then allows us to obtain (23), which finally completes the proof.

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[2] S. Loyka and A. Kouki, “On MIMO channel capacity, correlations, andkeyholes: Analysis of degenerate channels,” IEEE Trans. Commun.,vol. 50, no. 12, pp. 1886–1888, Dec. 2002.

[3] P. Almers, F. Tufvesson, and A. F. Molisch, “Keyhole effects in MIMOwireless channels-measurements and theory,” IEEE Trans. Wireless

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Part III

Part III

Multi-Keyhole Effect in MIMO AF Relay

Downlink Transmission with Space-Time Block

Codes

Part III is published as

T. Q. Duong, H. A. Suraweera, C. Yuen, and H.-J. Zepernick, “Multi-KeyholeEffect in MIMO AF Relay Downlink Transmission with Space-Time Block Codes,”in Proc. IEEE Global Communications Conference, Houston, TX, Dec. 2011.

Multi-Keyhole Effect in MIMO AF Relay

Downlink Transmission with Space-Time Block

Codes

Trung Q. Duong, Himal A. Suraweera, Chau Yuen,

and Hans-Jurgen Zepernick

Abstract

Multi-keyhole bridges the gap between single keyhole and full-scatteringmultiple-input multiple-output (MIMO) channels. In this paper, we there-fore investigate the multi-keyhole effect on the MIMO amplify-and-forward(AF) relay downlink transmission with orthogonal space-time block codes.In particular, we derive the analytical symbol error rate (SER) expressionfor the considered system with arbitrary number of keyholes. Moreover,SER approximations in the high SNR regime for several important specialscenarios of multi-keyhole channels are further derived. These asymptoticresults provide important insights into the impact of system parameterson the SER performance. Our analysis is confirmed by comparing withMonte-Carlo simulations.

1 Introduction

In recent years, a large body of research has studied the transmission of orthog-onal space-time block code (OSTBC) in relay networks to explore the spatialdiversity gain in a distributed manner (see, e.g., [1, 2, 3, 4] and the referencestherein). In particular, by incorporating the line-of-sight effect into the sec-ond hop, the symbol error rate (SER) over dual-hop Rayleigh-Rician fadingchannels has been investigated in [1]. The error probability of multiple-inputmultiple-output (MIMO) systems using OSTBCs and semi-blind amplify-and-forward (AF) relays over dual-hop Rayleigh fading channels has been presentedin [2]. For CSI-assisted AF relay, the bit error rate (BER) of MIMO systemwith OSTBC transmission has been presented in [3].

93

94 Part III

Although rich-scattering conditions have been widely employed in the exist-ing literature on MIMO, the assumption fails to model some practical scenarioswith lack of scattering objects. The later propagation phenomenon can be char-acterized as a keyhole or a pinhole MIMO channel. The keyhole channel hasbeen validated by conducting many measurement campaigns [5]. The keyholemodeling describes the most extreme case of channel fading, i.e., the channelmatrix is unit-rank regardless of the number of transmit/receive antennas. Asa consequence, keyhole effects substantially decrease the performance of MIMOsystems.

Despite of this important practical phenomenon, however, to the best ofthe authors’ knowledge, little is known on the effects of keyhole propagation indual-hop relay systems. Very recently, the effect of keyhole on the performanceof relay systems has been investigated in [6, 7, 8]. In [6, 7], the authors haveshown that using the decode-and-forward relay can mitigate the deleteriouseffect of keyhole. In [8], we adopted the same practical system model assumedin [6, 7] and investigated the effect of antenna correlation on the performanceof single keyhole MIMO AF relay systems. It has been shown that the antennacorrelation has no impact on the diversity order. In this paper, we take astep further and investigate the effects of multi-keyhole propagation for OSTBCtransmission in AF relay systems. In many practical scenarios, the degenerationeffect of poor-scattering environment can be more generally modeled as multi-keyhole channels. This generalization embraces both rich-scattering and keyholeconditions in which the rank of channel matrix now can vary from unit to full-rank [9, 10].

Our new contributions are as follows. We first characterize the end-to-endinstantaneous received signal-to-noise ratio (SNR), which allows us to obtainthe exact moment generating function (MGF) of end-to-end SNR. By utiliz-ing this result, we derive an analytical expression for the exact SER and anapproximate expression for a downlink system where an nS-antenna base sta-tion (source) communicates with an nD-antenna mobile station (destination)through the assistance of an nR-antenna relay, i.e., nS > min(nR, nD). The de-rived MGF expression is also useful to study additional important performancecriteria such as the outage probability and the ergodic capacity. We have alsopresented a high SNR analysis where both diversity and array gain are quantifiedin several special cases of interest. It is demonstrated that under the consid-ered downlink scenario, the diversity order is min(nK, nR) for a multi-keyholeMIMO/multiple-input single-output (MISO) channel, i.e., nD = 1, and nK isthe number of keyholes.

Notation: Vectors and matrices are denoted in lower case/upper case bold-

Multi-Keyhole Effect in MIMO AF Relay Downlink Transmission with STBC 95

face, respectively. IIIn represents the n × n identity matrix and ‖AAA‖F definesthe Frobenius norm of the matrix AAA. E . is the expectation operator. det(AAA)means the determinant of the matrix AAA. Kν (·) is the modified Bessel func-tion of the second kind [11, Eq. (8.432.3)] and B (·, ·) is the Beta function [11,Eq. (8.380.1)].


2.1 Protocol Description

We consider a dual-hop MIMO relay system with source, relay and destinationterminals having nS, nR and nD antennas respectively. Let HHH1 ∈ CnR×nS bethe channel matrix between the source and the relay, and HHH2 ∈ C

nD×nR bethe channel matrix between the relay and the destination. The channel gainsare assumed to be fixed during a block of Tc symbols and slowly changed overindependent blocks. The system under consideration is downlink in which thesource acts as a base station and the destination is a mobile station, i.e., nS >min(nR, nD). Moreover, we assume that the direct link does not exist in thissystem possibly due to high shadowing and large path loss effects.

An OSTBC is generated at the source before transmission over the dual-hop AF relay network. A sequence of information symbols x1, x2, ..., xN isselected from a constellation S with average transmit power per symbol Es,i.e., E

|xi|2

= Es. These symbols are then encoded into an OSTBC matrixdenoted by an Tc×nS transmission matrix GGG with the property that columns ofGGG are orthogonal. The input-output relationship for the communication of thesource-relay link is expressed as

YYY 1 =HHH1XXX +WWW 1, (1)

where the OSTBC transmission matrix at the source is defined as XXX = GGGT andWWW 1 denotes the additive white Gaussian noise (AWGN) matrix at the relay.The relay then multiplies the received signal YYY 1 with a fixed amplifying gain G,and retransmits the resulting signal to the destination. At the destination, theinput-output relationship is given by

YYY 2 =HHH2GYYY 1 +WWW 2, (2)

whereWWW 2 is the AWGN matrix at the destination. In this paper, without loss ofgenerality, we assume that the relay terminal operates in semi-blind mode and

96 Part III

consumes the same amount of power as the source, leading to the amplifyinggain G =

√

γ/[nR (1 + γ)], where γ = Es/N0 is the average SNR. The end-to-end instantaneous SNR is written as [1, 12]

γD = aγ

∥

∥

∥

∥

(

IIInD+G2HHH2HHH

†2

)−1/2

HHH

∥

∥

∥

∥

2

F

, (3)

where a = Tc/(NnS).

2.2 Channel Model

We consider a downlink cellular network where the source and the relay termi-nals are fixed base stations and can be installed at strategic locations by networkoperators. In contrast, the destination is a mobile station. In the consideredsystem model, it is reasonable to assume that the source-relay link enjoys arich-scattering environment yielding its corresponding channel matrix HHH1 tohave Rayleigh fading with entries CN (0, 1). On the other hand, we considera practical scenario in which the mobile station is located in a poor scatteringenvironment, e.g. due to mobility. In order to characterize this scenario, thechannel matrixHHH2 is assumed to undergo multi-keyhole fading with nK numberof keyholes. The multi-keyhole channel model extends the widely used singlekeyhole channel model (see e.g., [6, 7, 8]). In fact, the multi-keyhole channel canbe viewed as a generalized model which bridges the gap between single-keyholeand rich scattering MIMO channels. The multi-keyhole fading channel can bemathematically represented as

HHH2 =

nK∑

k=1

√σkhhhr,khhh

†t,k =HHHrAAAHHH

†t , (4)

where σk is the power gain of the k-th keyhole and AAA is the diagonal matrixwhose k-th diagonal element is

√σk. Moreover, HHHr = [hhhr,1, . . . ,hhhr,nK

] andHHHt = [hhht,1, . . . ,hhht,nK

] are mutually independent matrices. Note that, dependingon the keyhole number nK, the rank of matrixHHH2, can vary from one to full-rank.

3 Performance Analysis

3.1 The MGF of the end-to-end instantaneous SNR

In this subsection, we will derive the MGF of γD, which will be utilized toobtain the system performance. By the definition, the MGF is the inverse


Laplace transform of the probability density function (PDF), i.e., ΦγD(s) =

EγDexp(−sγD). As can be observed from (3), the MGF of γ2 can be expressed

as

ΦγD(s) = E

a

∥

∥

∥

∥

(

IIInD+G2HHH2HHH

†2

)−1/2

GHHH2HHH1

∥

∥

∥

∥

2

F

. (5)

Since HHH1 follows matrix-variate complex Gaussian distribution, by applyingresults from random matrix theory [1, 12], after some algebraic manipulations,we obtain

ΦγD(s) = E

det(

IIInD+G2HHH2HHH

†2

)

det(

IIInD+G2 (1 + sαΩ1)HHH2HHH

†2

)

nS

. (6)

From (6), and using the property that HHH2HHH2† is a positive-definite matrix, (6)

can be rewritten as

ΦγD(s) = EΛ

q∏

i=1

(

1 +G2λi1 +G2(1 + aγs)λi

)nS

, (7)

where λi, i = 1, . . . , q, are the q non-zero ordered eigenvalues of HHH2HHH†2 and

Λ = diag (λ1, . . . , λq). Averaging in (7) requires the joint PDF of λ1, . . . , λq forwhich a combined general expression is presented in the appendix.

3.2 Exact Symbol Error Rate

From (7), (36), and using Lemma 1 in [13], the MGF of γD can be readily derivedas

ΦγD(s) =

det(SSS)

K det(σj−1i )

, (8)

where K =∏qi=1 Γ(nD − i+ 1)Γ(nR − i+ 1) and SSS is an nK × nK matrix whose

entities are given by

SSSi,j =

σi−1j , 1 ≤ i ≤ t,∫∞0

2σnK−

nR+nD

2−1

j xnR+nD

2+i−t−q−1

×KnD−nR

(

2√

xσj

)

(

1 + G2aγxs1+G2x

)−nS

, t < i ≤ nK.

98 Part III

In this paper, we consider M -PSK modulated source symbols. Therefore, theaverage SER can be expressed as [14]

Pe =1

π

∫ π− πM

0

ΦγD

(

g

sin2 θ

)

dθ, (9)

where g = sin(

πM

)2. Unfortunately, to the best of our knowledge, (9) can not

be evaluated in closed-form. However, note that it is a convenient result andallows us to numerically evaluate the system’s average SER performance.

3.3 High SNR Symbol Error Rate

To provide further insight into the impact of multi-keyhole phenomenon onthe performance of the system, we will further scrutinize two important specialscenarios: i) Single keyhole MIMO/MIMO relay channels (nK = 1) and ii) Multi-keyhole MIMO/MISO relay channels (nD = 1). An asymptotic expression forΦγD

(s) for arbitrary combinations of nS, nR, nD, and nK appears prohibitivelycomplicated to obtain, if not impossible.

3.3.1 Single keyhole MIMO/MIMO Channels (nK = 1)

In this special scenario, ΦγD(s) given in (7) becomes

ΦγD(s) =

∫ ∞

0

2σnR+nD

2

Γ(nR)Γ(nD)x

nR+nD

2−1

× KnD−nR

(

2

√

x

σ

)(

1 +G2aγxs

1 +G2x

)−nS

. (10)

To approximate ΦγD(s) in the high SNR, we employ [15]

Kn(

2√z) (small z)

≈ − 1

2 ln (z) for n = 0,Γ(|n|)

2 z−|n|/2 for n 6= 0.(11)

• nR = nD: By substituting (11) into (10) and exchanging the variableγx = y, we have

ΦγD(s)

(large γ)≈ (σγ)−nR

Γ(nR)Γ(nD)

[∫ ∞

0

ln(σγ)ynR−1(

1 +G2asy)−nS

dy

−∫ ∞

0

ynR−1 ln(y)

(1 + aG2sy)nSdy

]

, (12)


where (12) is obtained by neglecting the small term in (10). By applyingthe results of [11, Eq. (3.194.3)] and [16, Eq. (2.6.4.7)] with the conditionthat nS > nR, the MGF of γD for the case nR = nD can be expressed as

ΦγD(s)

(large γ)≈ (aγs)

−nRΓ(nS − nR)

Γ(nS)Γ(nR)

× [ln (aγs) − ψ(nR) + ψ(nS − nR)] , (13)

where a1 = σG2a.

• nR 6= nD: Similarly, we can find the MGF of γD as

ΦγD(s)

(large γ)≈ (σγ)−m1

Γ(nR)Γ(nD)

∫ ∞

0

ym1−1

(1 + aG2sy)nSdy, (14)

where m1 = min(nR, nD). Note that the integral in (14) converges whennS > min(nR, nD) which then yields

ΦγD(s)

(large γ)≈ Γ(nS −m1)Γ(m1)Γ(|nD − nR|)

Γ(nS)Γ(nR)Γ(nD) (a1γs)m1

. (15)

By combining the two cases, the asymptotic SER can be expressed as

Pe

(large γ)≈ Ξ1

Γ(nS −m1)Γ(m1)Γ(|nD − nR|)Γ(nS)Γ(nR)Γ(nD) (a1γs)

m1, (16)

with the condition that nS > min(nR, nD) and Ξ1 is defined, for example withM -PSK modulation as

Ξ1 =

1π

∫ π− πM

0

(

g

sin2 θ

)−nR[

ln(

γa1g

sin2 θ

)

−ψ(nR) + ψ(nS − nR)]

dθ, for nR = nD,Γ(|nD−nR|)

π

∫ π− πM

0

(

sin2 θg

)m1

dθ, for nR 6= nD.

3.3.2 Multi-keyhole MIMO/MISO Channels (nD = 1)

The case of multi-keyhole MIMO/MISO system setup is reasonable in the con-text of current cellular networks where for e.g., due to space constraints mobilestation is only equipped with a single antenna. In this special scenario, i.e.,nD = 1, the elements of matrix SSS in (8) are represented as SSSij = σi−1

j for

100 Part III

i = 1, . . . , nK − 1 and the last row of SSS is given by

SSSnKj =

∫ ∞

0

2σnK−

nR+1

2−1

j xnR+1

2

× KnR−1

(

2

√

x

σj

)(

1 +G2aγxs

1 +G2x

)−nS

dx. (17)

Due to the multi-linear property of the determinant, to calculate det(SSS) forlarge γ, we use the series representation of the Bessel function as

Kν (z) =

ν−1∑

k=0

Γ(ν − k)

Γ(k + 1)

(−1)k

2

(z

2

)−ν+2k

+

(−1)ν+1∞∑

k=0

( z2 )ν+2k[

ln( z2 ) − ψ(k+1)2 − ψ(ν+k+1)

2

]

Γ(k + 1)Γ(ν + k + 1). (18)

From (17), (18), and after several calculations together with the help of [11,Eq. (3.194.3)] and [16, Eq. (2.6.4.7)], we can express the entries of the last rowof matrix SSS as two terms I1 and I2, i.e, SSSnKj = I1 + I2, respectively shown asfollows:

I1 ≈nR−2∑

k=0

(−1)kΓ(nR − k − 1)Γ(nS − k − 1)σnK−k−2j

Γ(nS)(G2aγs)k+1

I2 ≈ (−1)nR+1∞∑

k=0

σnK−nR−k−1j B (nR + k, nS − nR − k)

Γ(k + 1)Γ(nR + k)(G2aγs)k+1

×[

ln(G2aγs) + ψ(nS − nR − k) + ψ(k + 1)]

. (19)

To asymptotically approximate the determinant of matrix SSS, we consider twoseparate cases: i) nR > nK and ii) nR ≤ nK When nR > nK: The minimumexponent of γ to make det(SSS) non-zero in I1 is k = nK−1 and in I2 is k = nR−1.Hence, I2 can be neglected as compared to I1, which results in

ΦγD(s)

(large γ)≈ Γ(nR − nK)Γ(nS − nK) det(SSS1)

KΓ(nS)(G2aγs)nK det(σj−1i )

, (20)

where SSS1 is an nK × nK matrix whose entities are

SSS1i,j =

σi−1j , 1 ≤ i ≤ nK − 1,

(−1)nK−1σ−1j , i = nK.


Since det(σj−1i ) =

nK∏

1≤i<j≤nK

(σj − σi) and

det(SSS1) =

nK∏

k=1

σ−1k

nK∏

1≤i<j≤nK

(σj − σi),

we can rewrite (20) as

ΦγD(s)

(large γ)≈ K−1Γ(nR − nK)Γ(nS − nK)

Γ(nS)∏nK

k=1 σk(G2aγs)nK

. (21)

When nR ≤ nK: In this case, only I2 is taken into account and the minimumexponent of γ in I2 can be selected as k = 0, which yields

ΦγD(s)

(large γ)≈ Γ(nS − nR) det(SSS2)

KΓ(nS)(G2aγs)nR det(σj−1i )

, (22)

where SSS2 is an nK × nK matrix whose entities are as

SSS2i,j =

σi−1j , 1 ≤ i ≤ nK − 1,

(−1)nR+1σnK−nR−1j [ln(G

2aγsσj

)

+ψ(nS − nR) + ψ(1)], i = nK.

Additionally, we can compute det(SSS2) as

det(SSS2) = (−1)nR+nK [ln(G2aγs) + ψ(nS − nK) + ψ(1)]

× det(SSS1) + det(SSS3), (23)

where SSS3 is an nK × nK matrix whose entities are as

SSS3i,j =

σi−1j , 1 ≤ i ≤ nK − 1,

(−1)nR+1σnR−nK−1j ln(σj), i = nK.

From (22) and (23), ΦγD(s) can be asymptotically approximated as

ΦγD(s)

(large γ)≈ K−1Γ(nS − nR)

Γ(nS)(G2aγs)nR

[

det(SSS3)

det(σi−1j )

+ (−1)nR+nK

× [ln(G2aγs) + ψ(nS − nR) + ψ(1)]

nK∏

k=1

σ−1k

]

. (24)

102 Part III

0 5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

nK=1, 2, 3

Analysis Simulation

8-PSK

Multikeyhole (4,3,2)-MIMO AF relay

SNR (dB)

Sym

bol E

rror

Rat

e

Figure 1: SER of (4, 3, 2)-MIMO AF relay systems with multi-keyhole versusSNR for 8-PSK modulation and nK = 1, 2, 3.

By combining (21), (24) with (9), the average SER can be asymptotically ap-proximated as

Pe

(large γ)≈ K−1Γ(nS − min(nR, nK))

πΓ(nS)(G2agγ)min(nR,nK)Ξ2, (25)

where Ξ2 is a constant and given by

Ξ2 =

Γ(nR − nK)∫ π− π

M

0

(

sin2 θ)nK

dθ, for nR > nK,∫ π− π

M

0 det(SSS3)

det(σi−1

j )+ (−1)nR+nK [ln(G

2aγg

sin2 θ)

+ψ(nS − nR) + ψ(1)]∏nK

k=1 σ−1k dθ, for nR ≤ nK.

From (25), we see that the diversity order for multi-keyhole MIMO/MISO relaysystems is equal to min(nK, nR).


4 Numerical Results

In this section, we provide numerical results for some representative scenariosto validate our analysis in previous section. For notational brevity, we define aMIMO AF relay system as (nS, nR, nD) and all results are shown for 8-PSK mod-ulation. For the OSTBC transmission, we apply the general approach presentedin [17] for arbitrary nS, for e.g., when nS=4 and 5, the code rate Rc = 3/4 and1/2, respectively.

Fig. 1 displays the SER performance versus SNR for (4, 3, 2)-MIMO AF relaysystem when nK = 1, 2, 3 and σj3

j=1 = 0.5, 0.3, 0.2. As can be observed fromFig. 1, the analytical results show a good agreement with simulations. We seethat the performance improves with the number of keyholes nK.

0 5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

nK=1

Case 2

Case 3

Case 1

Case 1: (5,3,2)Case 2: (5,3,3)Case 3: (5,4,3)

Analytical Asymptotic

SNR (dB)

Sym

bol E

rror

Rat

e

Figure 2: SER of (5, 3, 2), (5, 3, 3), and (5, 4, 3)-MIMO AF relay systems withkeyhole versus SNR for 8-PSK modulation.

Fig. 2 compares the asymptotic and analytical SER for keyhole channels, i.e.,nK = 1 and σ = 1. Results are shown for different MIMO AF relay systems,i.e., Case 1: (5, 3, 2), Case 2: (5, 3, 3), Case 3: (5, 4, 3). The asymptotic curvesprecisely converge to analytical ones in the high SNR regime. We see that whenincreasing nR from three to four, the diversity gain does not increase as the SER

104 Part III

5 10 15 20 25 30 3510-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Case 7

Case 5

Case 6

Case 4

Case 4: (5,4,1) & nK=2

Case 5: (5,2,1) & nK=4

Case 6: (5,3,1) & nK=4

Case 7: (5,6,1) & nK=4

Analytical Asymptotic

SNR (dB)

Sym

bol E

rror

Rat

e

Figure 3: SER of (5, nR, nD)-MIMO AF relay systems with keyhole versus SNRfor 8-PSK modulation and nD = 1.

curves for Case 2 and Case 3 are parallel. This observation is inline with theour derivation in previous section as the diversity gain depends on min(nR, nD).

Fig. 3 shows the SER performance for multi-keyhole MIMO/MISO AF relaysystems. Results are shown for various antenna configurations and number ofkeyholes, i.e., Case 4: nR = 4 and nK = 2, Case 5: nR = 2 and nK = 4, Case6: nR = 3 and nK = 4, Case 7: nR = 6 and nK = 4. The number of antenna atsource nS = 5 and multi-keyhole powers are σj4

j=1 = 0.4, 0.3, 0.2, 0.1. Wecan observe that the asymptotic approximations are accurate in the high SNRregime and the SER performance is determined by the minimum of nK and nR.Specifically, among the four cases, Case 7 surpasses Case 4, 5, and 6 since Case7 has the highest value of min(nK, nR).

Finally, to further understanding the effect of multi-keyhole on the SER per-formance, Fig. 4 displays the performance of (5, 3, 2)-MIMO AF relay systemswith various nK. For comparison, the case of MIMO AF relay channel whereboth hops undergo Rayleigh fading is also plotted in Fig. 4. It is observedthat as nK ≤ 2 the diversity gain does not increase as expected. Moreover, theperformance is significantly improved when nK is increased from 1 to 8, while


5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

nK=2,4,8,12

(5,3,2)-MIMO AF relay8-PSK

Keyhole Multi-keyhole Rayleigh

SNR (dB)

Sym

bol E

rror

Rat

e

Figure 4: SER of (5, 3, 2)-MIMO AF relay systems with multi-keyhole versusSNR for 8-PSK modulation and nK = 1, 2, 4, 8, 12.

additional keyholes, (nK > 8), has a diminishing impact on reducing the SER.When the number of keyholes increases in the limit nK → ∞, the performanceapproaches to that of Rayleigh MIMO AF relay channels [12].

5 Conclusion

In this paper, we have investigated the multi-keyhole effect on the SER perfor-mance of MIMO AF relay systems by deriving the analytical SER expression.The analytical results enable us to investigate the multi-keyhole effect whichencompasses a variety of MIMO fading channels from keyhole to full-scatteringenvironment. We have also obtained the asymptotic approximation of SERfor several important cases including single keyhole MIMO/MIMO AF relaysystems, multi-keyhole MIMO/MISO AF relay systems. Specifically, we haveshown that for multi-keyhole MIMO/MISO AF relay channels (nD = 1), thediversity gain is solely determined by the minimum among the number of key-holes and the number of antennas at the relay, i.e., min(nK, nR). The tightly

106 Part III

converging asymptotic results provide insights into the effect of multi-keyholeand system’s configuration on the SER performance.

Appendix

In this appendix, we derive the PDF of λλλ= [λ1, λ2, . . .]. Due to symmetry of thechannel, we only deal with the case of nR > nD.

The nD > nK Case

Let 0 ≤ λ1 ≤ . . . ≤ λnK≤ ∞ be the eigenvalues of HHH†

2HHH2 with HHH2 ∈ CnD×nR .Define Σ = AA† and BBB = A†H†

rHrA. Hence, we can write

HHH†2HHH2 = H

†tBBBHt. (26)

Let Dord = (x1, . . . , xnK) be the ordered eigenvalues of BBB with 0 ≤ x1 ≤ . . . ≤

xnK≤ ∞. Conditioned on Dord, the joint PDF of the non-zero eigenvalues of

HHH†2HHH2 is given by the joint PDF of the eigenvalues of BBB

12 HtH

†tBBB

12 . This has

the same form as A†H†rHrA and so the PDF comes from [18, Eq. (42)] as

f(λ1, . . . , λnK|x1, . . . , xnK

)

=

∏nK

i=1 λnR−nK

i det(

λnK−ji

)

det

(

e− λi

xj

)

∏nK

i=1 Γ(nR − i+ 1) det (BBB)nR∏nK

ℓ<k

(

1xk

− 1xℓ

) . (27)

Therefore,

f(λ) = Cmk1

∫

Dord

∏nK

i=1 xnD−nK

i

(∏nK

i=1 xi)nR∏nK

ℓ<k

(

1xk

− 1xℓ

)

× det(

qnK−ji

)

det(

e− qi

σj

)

det

(

e− λi

xj

)

dx1 . . . dxnK, (28)

where σi are the diagonal elements of Σ and

Cmk1 =det (Σ)

−nD∏nK

i=1 λnR−nK

i det(

λnK−ji

)

∏nK

i=1(nR − i)!(nD − i)!∏nK

ℓ<k

(

1σk

− 1σℓ

) . (29)


Using Corollary 2 in [19], we obtain

f(λ) =2nK

∏nK

i=1 λnR−nK

i det(

λnK−ji

)

∏nK

i=1(nR − i)!(nD − i)! det (Σ)nD∏nK

ℓ<k

(

1σk

− 1σℓ

)

× det

(

(λiσj)nD−nR

2 KnD−nR

(

2

√

λiσj

))

, (30)

where (30) is obtained from [11, Eq. (3.471.9)].

The nD ≤ nK Case

When nR ≥ nK ≥ nD or nR ≤ nK ≥ nD based on [20, Eq. (25)], we have jointordered eigenvalues x1 ≥ x2 ≥ · · · ≥ xnD

as

f(x1, . . . , xnD) =

det(

xj−1i

)

det (V )

∏nD

i=1 Γ(nD − i+ 1) det(

σj−1i

) , (31)

where matrix V is defined as

V i,j =

σj−1i , 1 < j ≤ nK − nD,

σnK−nD−1i e

−xj−nK+nD

σi , nK − nD < j ≤ nK.(32)

for i, j = 1, . . . , nK. From [18, Eq. (42)], we have

f(λ) = Cmk2

∫

Dord

det

(

e− λi

xj

)

× det(V )

nD∏

i=1

xnD−nR−1i dx1 . . . dxnD

, (33)

where Cmk2 is given by

Cmk2 =

∏nD

i=1 λnR−nD

i det(

λj−1i

)

∏nD

i=1 Γ(nD − i+ 1)Γ(nR − i+ 1) det(

σj−1i

) .

Using [13, Lemma 2], the integral in (33), can be solved as

I = det(φφφ), (34)

108 Part III

where φ is the nK × nK matrix whose entries are given by

φφφi,j =

σj−1i , 1 < j ≤ nK − nD,

σnK−nD−1i

∫∞0 xnD−nR−1

×e−λj−nK+nD

x− x

σi dx, nK − nD < j ≤ nK.

Pulling everything together, we have the following expression for the joint or-dered eigenvalue given by

f(λ) =Γ(nD + 1)

∏nD

i=1 λnR−nD

i det(

λj−1i

)

det(φφφ)

∏nD

i=1 Γ(nD − i+ 1)Γ(nR − i+ 1) det(

σj−1i

) . (35)

Finally, by combining (30) and (35), we can obtain the PDF of λλλ for the generalcase as

f(λ) =

∏qi=1 λ

nR−qi det

(

λj−1i

)

det(ΩΩΩ)∏qi=1 Γ(nD − i+ 1)Γ(nR − i+ 1) det(σj−1

i )nK

, (36)

where q = min(nD, nK) and ΩΩΩ is an nK × nK matrix with entries

ΩΩΩi,j =

σj−1i , 1 ≤ j ≤ t,

σnK−nD−1i 2(λj−tσi)

nD−nR2

×KnD−nR

(

2√

λj−t

σi

)

, t < j ≤ nK.

where t = max(0, nK − nD).

References

[1] T. Q. Duong, H. Shin, and E.-K. Hong, “Effect of line-of-sight on dual-hopnonregenerative relay wireless communications,” in Proc. IEEE VTC Fall

2007, Baltimore, MD, Sep. 2007, pp. 571–575.

[2] Y. Song, H. Shin, and E.-K. Hong, “MIMO cooperative diversity withscalar-gain amplify-and-forward relaying,” IEEE Trans. Commun., vol. 57,no. 7, pp. 1932–1938, Jul. 2009.



[4] S. Chen, W. Wang, and X. Zhang, “Performance analysis of OSTBC trans-mission in amplify-and-forward cooperative relay networks,” IEEE Trans.

Veh. Technol., vol. 59, no. 1, pp. 105–113, Jan. 2010.

[5] P. Almers, F. Tufvesson, and A. F. Molisch, “Keyhole effects in MIMOwireless channels-measurements and theory,” IEEE Trans. Wireless Com-

mun., vol. 5, no. 12, pp. 3596–3604, Dec. 2006.

[6] O. Souihli and T. Ohtsuki, “Cooperative diversity can mitigate keyhole ef-fects in wireless MIMO systems,” in Proc. IEEE GLOBECOM 2009, Hon-olulu, HI, Nov./Dec. 2009, pp. 1–6.

[7] ——, “The MIMO relay channel in the presence of keyhole effects,” in Proc.

IEEE ICC 2010, Cape Town, South Africa, May 2010.

[8] T. Q. Duong, H. A. Suraweera, T. A. Tsiftsis, H.-J. Zepernick, and A. Nal-lanathan, “OSTBC transmission in MIMO AF relay systems with keyholeand spatial correlation effects,” in Proc. IEEE ICC 2011, Kyoto, Japan,Jun. 2010, to appear.

[9] G. Levin and S. Loyka, “Multi-keyhole MIMO channels: Asymptotic analy-sis of outage capacity,” in Proc. IEEE Int. Symp. on Inform. Theory (ISIT),Seattle, WA, Jul. 2006, pp. 1305–1309.

[10] C. Zhong, S. Jin, K.-K. Wong, and M. R. McKay, “Outage analysis for op-timal beamforming MIMO systems in multikeyhole channels,” IEEE Trans.

Signal Process., vol. 58, no. 3, pp. 1451–1456, Mar. 2010.

[11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,6th ed. San Diego, CA: Academic, 2000.

[12] P. Dharmawansa, M. R. McKay, and R. K. Mallik, “Analytical performanceof amplify-and-forward MIMO relaying with orthogonal space–time blockcodes,” IEEE Trans. Commun., vol. 58, no. 7, pp. 2147–2158, Jul. 2010.

[13] H. Shin, M. Z. Win, J. Lee, and M. Chiani, “On the capacity of doublycorrelated MIMO channels,” IEEE Trans. Inf. Theory, vol. 5, no. 8, pp.2253–2265, Aug. 2006.

[14] M. Abramowitz and I. A. Stegun, Handbook of mathematical functions wih

Formulas, Graphs, and Mathematical Tables. Washington, DC: U. S. Dept.Commerce, 1970.

110 Part III

[15] ——, Handbook of mathematical functions. New York: Dover PublicationsInc, 1974.

[16] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series.New York: Gordon and Breach Science, 1986, vol. 1.

[17] O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space–timeblock codes for complex signal constellations,” IEEE Trans. Inf. Theory,vol. 48, no. 2, pp. 384–395, Feb. 2002.

[18] M. Kang and M.-S. Alouini, “Capacity of correlated MIMO Rayleigh chan-nels,” IEEE Trans. Wireless Commun., vol. 5, no. 1, pp. 143–155, Jan.2006.

[19] M. Chiani, M. Z. Win, and A. Zanella, “On the capacity of spatially corre-lated MIMO Rayleigh-fading channels,” IEEE Trans. Inf. Theory, vol. 49,no. 10, pp. 2363–2371, Oct. 2003.

[20] P. J. Smith, S. Roy, and M. Shafi, “Capacity of MIMO systems with semi-correlated flat fading,” IEEE Trans. Inf. Theory, vol. 49, no. 49, pp. 2781–2788, Oct. 2003.

Part IV

Part IV

Distributed Space-Time Coding in Two-Way

Fixed Gain Relay Networks over Nakagami-m

Fading

Part IV is published as

T. Q. Duong, H. Q. Ngo, H.-J. Zepernick, A. Nallanathan, “Distributed Space-Time Coding in Two-Way Fixed Gain Relay Networks over Nakagami-m FadingNetworks,” in Proc. IEEE International Communications Conference, Ottawa,Canada, June 2012.

Based on

T. Q. Duong, C. Yuen, H.-J. Zepernick, X. Lei, “Average Sum-Rate of Dis-tributed Alamouti Space-Time Scheme in Two-Way Amplify-and-Forward Re-lay Networks,” in Proc. IEEE Global Communications Conference Workshop,Miami, FL, Dec. 2010.

Distributed Space-Time Coding in Two-Way

Fixed Gain Relay Networks over Nakagami-m

Fading

Trung Q. Duong, Hien Quoc Ngo, Hans-Jurgen Zepernick,

and Arumugam Nallanathan

Abstract

The distributed Alamouti space-time code in two-way fixed gain amplify-and-forward (AF) relay is proposed in this paper. In particular, closed-form expressions for approximated ergodic sum-rate and exact pairwiseerror probability (PWEP) are derived for Nakagami-m fading channels.To reveal further insights into array and diversity gains, an asymptoticPWEP is also obtained. Finally, numerical results are provided to cor-roborate the proposed theoretical analysis.

1 Introduction

Space-time codes (STC) for two-way relay networks have been recently consid-ered for efficient transmission to achieve full diversity order and recover the lossin spectral efficiency inflicted by dual-hop transmission [1]. The performanceof distributed space-time codes in two-way relay networks has drawn great at-tention [2, 3, 4, 5, 6, 7]. Specifically, the lower and upper bounds of averagesum-rate for Alamouti space-time codes (ASTCs), where each source equippedwith two antennas transmits Alamouti space-time code through the assistanceof a channel state information (CSI)-assisted amplify-and-forward (AF) relay,have been derived in [2, 3]. Then, the upper and lower bounds for error rateperformance for ASTC have been presented in [4]. The upper bound of symbolerror probability for the same system has been presented in [5]. It has beenshown in [5] that second diversity order is obtained for ASTC. In addition, theselection mechanism for two-way ASTC transmission has been discussed in [6].

115

116 Part IV

The above mentioned works, i.e., [2, 3, 4, 5, 6], have extended the concept ofASTC considered for one-way AF relay networks [8] to its two-way counterpart.In other words, the two sources must deploy multiple antennas to achieve thefull diversity gain. However, multiple antenna deployment is limited by practi-cal space constraint at portable devices. To overcome this drawback, an ASTCtransmission scheme for two-way AF relay networks has been proposed for allsingle antenna nodes [7]. With this strategy, the two single-antenna sources co-operate in a distributed fashion to generate ASTC, namely distributed Alamoutispace-time code (DASTC). In particular, a closed-form expression of approxi-mated ergodic sum-rate has been derived for Rayleigh fading channels [7]. Sofar, the performance of ASTC/DASTC for two-way AF relay networks has onlybeen investigated for Rayleigh fading channels (see, e.g., [2, 3, 4, 5, 6, 7]). In theavailable technical literature, to the best of the authors’ knowledge, there hasbeen no previously published work considering ASTC/DASTC two-way trans-mission over Nakagami-m fading channels. Despite the great interest, all of theabove mentioned works [2, 3, 4, 5, 6, 7] only concentrated on CSI-assisted AFrelay.

Inspired by the above discussion, in this paper, the fixed gain AF relay for thefirst time has been applied to DASTC two-way transmission. The closed-formexpression for ergodic sum-rate and pairwise error probability (PWEP) overNakagami-m has been derived. The two-way system has been shown to surpassthe commonly considered one-way counterpart by one nats/s/Hz, which is aremarkable improvement in spectral efficiency knowing that the contemporarywireless system can support up to 2-3 nats/s/Hz. Moreover, the asymptoticPWEP revealing the array and diversity gains are also derived. Finally, theanalysis is validated by comparing with Monte Carlo simulations.

2 System Model

Consider a two-way relay networks where two terminals T1 and T2 exchangetheir information with the help of a fixed gain AF relay node R. The Alamoutischeme is applied to this two-way model. We assume that the flat fading chan-nels of all links remain constant for a block spanning over at least six successivesymbols intervals and vary independently for every block. Under a half-duplexconstraint, the communications occur in three phases as shown in Fig. 1. Inthe first phase, T1 and T2 simultaneously transmit their information to R. The

DSTC in Two-Way Fixed Gain Relay Networks over Nakagami-m Fading 117

s1

x1

[ ]s s=s1 1 2

[ ]x x=x1 1 2

Gyx

2

x x = − x

2 2 1

s2

Gy

s s = − s

2 2 1

h1

h2

h1

h0

h2

h0

Figure 1: Two-way relay channels with DASTC.

received signal at R is given by

yyyR = h1sss1 + h2xxx1 +nnnR (1)

where sss1 = [s1 s2] and xxx1 = [x1 x2] are respectively the first rows of the

Alamouti code matrix transmitted from T1 and T2 with E

|sk|2

= p1 and

E

|xk|2

= p2, k = 1, 2, E · denotes the statistical expectation. In (1),

hk, the channel coefficient from Tk to the relay R, follows complex Gaussiandistribution with zero mean and variance Ωk, and nnnR is a 1 × 2 additive whiteGaussian noise (AWGN) vector whose elements are Gaussian distributed withzero mean and variance N0. We assume the channel reciprocity for hk as in [1].

In the second phase, R amplifies the received signal with fixed relaying gainG that satisfies the average power constraint, pR, and forwards this signal toT2. At the same time, T1 sends the second row of the Alamouti code matrixsss2 = [−s∗2 s∗1] to T2. The received signal at T2 is given by

yyy2 = h2GyyyR + h0sss2 +nnn2 (2)

where h0 is the channel coefficient for the link between T1 and T2, and nnn2 isa 1 × 2 AWGN vector at T2. We assume that the elements of nnn2 are Gaussiandistributed with zero mean and variance N0.

In the third phase, T2 and R are in the transmit mode, whereas T1 switchesto the listening mode. T2 sends its second row of the Alamouti code matrixxxx2 = [−x∗2 x∗1], while R sends its amplified version GyyyR to T1. The receivedsignal at T1 is

yyy1 = h1GyyyR + h0xxx2 + nnn1 (3)

where nnn1 is a 1×2 AWGN vector at T1 whose elements are Gaussian distributedwith zero mean and variance N0.

118 Part IV

We assume that Tk has perfect CSI of all links. Since Tk knows its owntransmitted signal, it can remove self-interference prior to decoding. Thus, thereceived signals at T1 and T2 in the second and third phases are respectivelygiven by

yyy1 = hhhXXX + zzz1, yyy2 = hhhSSS + zzz2 (4)

where hhh = [Gh1h2 h0], zzzk = GhknnnR + nnnk, and SSS =

[

s1 s2−s∗2 s∗1

]

, XXX =[

x1 x2

−x∗2 x∗1

]

are Alamouti space-time codes.

3 Ergodic Sum-Rate and PWEP Analysis

In this section, we derive the closed-form expressions for the ergodic sum-rateand PWEP of the above two-way fixed gain AF relaying with DASTC. Assumingthat all communication links are subject to independent distributed Nakagami-m fading. Let γi be the squared magnitude of hi, i.e., γi , |hi|2, i = 0, 1, 2.

Then γi is Gamma distributed, namely, γi ∼ G(

mi,γi

mi

)

, whose probability

density function (PDF) is given by

fγi(γi) =

mmi

i γmi−1i

Γ(mi)γmi

i

exp

(

−miγi

γi

)

, i = 0, 1, 2 (5)

where Γ (z) is gamma function [9, eq. (8.310.1)], mi is a fading severity param-eter, and γi is the average of γi.

3.1 Ergodic Sum-Rate

From (4), with the perfect CSI at T1 and T2, the ergodic sum-rate (nat/s/Hz)is given by

Csum = R1 +R2 (6)


where Rk is the ergodic rate of link Lk, i.e., communication from Tk to Tℓ withk 6= ℓ, represented as

Rk =1

3E

ln det

III2 +pkhhh

†hhh

N0

(

1 + α |h1|2 |h2|2 / |hk|2)

=1

3E

ln

(

1 +pk

N0

αγ1γ2 + γ0

αγ1γ2/γk + 1

)

(7)

where α , G2 = (γ1 + γ2 +1/γ)−1 and γ = Pk/N0 is the average SNR. Withoutloss of generality, let us assume p1 = p2 = p, which results in γ = p/N0. Notethat pre-factor 1/3 is due to the use of six time slots (each phase spends two timeslots) for the exchange of information consisting of two symbols per terminal1.As a result, the average sum-rate of the considered two-way relay system canbe written as

Csum =1

3E

ln[(

1 + γ αγ1γ2+γ0

αγ2+1

)(

1 + γ αγ1γ2+γ0

αγ1+1

)]

≈ 1

3E

ln

[

(αγγ1γ2)2

(αγ1 + 1)(αγ2 + 1)

(

1 +γ0

αγ1γ2

)2]

(8)

=2

3E ln(αγγ1γ2)︸︷︷︸

I1

+2

3E

ln

(

1 +γ0

αγ1γ2

)

︸︷︷︸

I2

− 1

3E ln (1 + αγ1)︸︷︷︸

J1

−1

3E ln (1 + αγ2)︸︷︷︸

J2

(9)

The above approximation in (8) is obtained by neglecting the unity term whichleads to a lower bound on the ergodic rate for link Lk. This bound is remarkablytight, particularly in the medium to high SNR regime. By using this approx-imation, we get a closed-form expression for ergodic sum-rate in the followingtheorem.

Theorem 1 The ergodic sum-rate of DASTC in two-way fixed gain AF relaying

1For ASTC, e.g. [2, 3, 4, 5, 6], the pre-factor 1/2 is used instead at the expense of multipleantenna deployment at T1 and T2.

120 Part IV

networks is given by

Csum ≈ 2

3

[

ln

(

γαγ1γ2

m1m2

)

+ ψ(m1) + ψ(m2)

]

+2

3Γ(m0)

× 1

Γ(m1)Γ(m2)G3,3

4,3

(

αm0γ1γ2

γ0m1m2

∣

∣

∣

∣

1 −m2, 1 −m1, 0, 1m0, 0, 0

)

− 1

3

2∑

n=1

1

Γ(mn)G1,3

3,2

(

αγn

mn

∣

∣

∣

∣

1 −mn, 1, 11, 0

)

(10)

where ψ(x) is the Euler psi function [9, Eq. (8.360.1)] and Gm,np,q [·] is Meijer’s

G-function [10, Eq. (8.2.1.1)].

Proof See Appendix A.

3.2 Pairwise Error Probability

In this section, we derive the closed-form expression of PWEP for each link Lk.Since the analysis is the same for links L1 and L2, we consider link L1, i.e., thecommunications T1 → T2. For a given CSI hhh at T2, yyy2|hhh is a Gaussian random

vector with mean hhhSSS and covariance matrix N0

(

G2 |h2|2 + 1)

III2. By using the

maximum-likelihood (ML) detection, the estimate of the transmitted codewordis given by

SSS = argminSSS

‖yyy2 − hhhSSS‖2F (11)

where the minimization is performed over all possible Alamouti codeword ma-trices SSS. We are interested in the probability that T2 decodes codeword EEEinstead of codeword SSS. This probability is known as PWEP. Conditioned onthe channel realization hhh, the PWEP of link L1 is given by

P (SSS → EEE|hhh) = Q

√

√

√

√

‖hhh (SSS −EEE)‖2F

2N0

(

G2 |h2|2 + 1)

= Q

√

√

√

√

p1d2E ‖hhh‖2

F

2N0

(

G2 |h2|2 + 1)

(12)

where Q (x) , 1π

∫ π/2

0exp

(

−x2

2 sin2 θ

)

dθ is the Gaussian Q-function, and the

last equality follows the fact that (SSS −EEE) (SSS −EEE)†

= p1d2EIII2, with d2

E =


|s1 − e1|2 + |s2 − e2|2. By averaging the conditional PWEP of (12) over allchannel realizations, we obtain the unconditional PWEP as

P (SSS → EEE) =1

π

∫ π/2

0

Mη1

p1d2E

4N0 sin2 θ

dθ (13)

where Mγ s , Eγ exp (−sγ) is the moment-generating function (MGF) ofthe random variable γ, and

η1 ,‖hhh‖2

F

G2 |h2|2 + 1=αγ1γ2 + γ0

αγ2 + 1(14)

By using the PDF of γi in (5), we derive the MGF of η1 and hence, obtain theclosed-form solution for the PWEP as the following theorem.

Theorem 2 For mi is a positive integer, PWER for link L1 is given by (13)where

Mη1s =

m0+m1∑

k=0

a0a1γm0+m1−k2 Γ (m0 +m1 +m2 − k)

bm0

0 bm1

1 αk−m0−m1Γ (m2)mm0+m1−k2

× F 1:10:0

[

m0 +m1 +m2 − k : m0 ;m1 ;- : - ; - ;

− c0γ2

b0m2,− c1γ2

b1m2

]

(15)

where ai =m

mii

γi, i = 0, 1, b0 = s+ m0

γ0

, b1 = m1

γ1

, c0 = m0αγ0

, c1 = sα+ m1αγ1

, and

Fm:np:q [·] is the Kampe de Feriet’s function.


The diversity order for the link Lk is defined as the slope of the error probability(e.g., PWEP) curve plotted on a loglog scale in high SNR regime, i.e.,

Dk = limγ→∞

− logP (SSS → EEE)

log (γ). (16)

From (31), by neglecting the small terms, we obtain the approximated expressionfor MGF of η1 as

Mη1s SNR1→∞≈ a0a1a2

Γ(m2)αm1sm0+m1

×∫ ∞

0

zm2−m1−1 (αz + 1)m0+m1 e−

m2γ2

zdz (17)

122 Part IV

0 5 10 15 20 25 300

1

2

3

4

m0=m1=m2=2

Two-Way (Analysis) Two-way (Simulation) One-Way (Analysis) One-way (Simulation)

SNR (dB)

Ergo

dic

Cap

acity

Figure 2: Ergodic capacity of two-way DASTC. Results are shown for γℓ2ℓ=0 =

0.5.

It is important to note that (17) only converges when m2 ≥ m1. Then, applyingthe binomial theorem, we obtain

Mη1s γ→∞≈ a0a1

Γ(m2)sm0+m1

m0+m1∑

k=0

(

m0 +m1

k

)

× Γ(m2 −m1 + k)

(

m2

αγ2

)m1−k

(18)

By substituting (18) into (13), the asymptotic expression for PWEP is shownas

P (SSS → EEE)γ→∞≈ a0a1Ξ

Γ(m2)(

d2E

4

)m0+m1

γm0+m1

(19)

where Ξ is a constant defined as

Ξ = ǫ

m0+m1∑

k=0

(

m0 +m1

k

)

Γ(m2 −m1 + k)

(

m2

αγ2

)m1−k

(20)


0 5 10 15 20 25 3010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Scenario 1: m0=m

1=m

2=3

Scenario 2: m0=m

1=m

2=2

Scenario 3: m0=m

1=m

2=1

Scenario 1, 2, 3

Analysis Simulation

SNR (dB)

Pair

wis

e Er

ror P

roba

bilit

y BPSKd

E

2=4

Figure 3: PWEP of two-way DASTC. Results are shown for γℓ2ℓ=0 = 1/16.

with ǫ = 1π

∫ π/2

0 (sin θ)2(m0+m1)dθ. From (19), we can observe that the diversityorder is m0 +m1 for the case m2 ≥ m1. Similarly following the same approach,we can obtain the diversity order asm0+m2 for the case m1 > m2. In summary,the achievable diversity order for link Lk of the considered system is

Dk = m0 + min(m1,m2) (21)

4 Numerical Results

In this section, we provide the numerical results to illustrate the effect of two-way DASTC transmission on the system performance. We assume that theAlamouti code is constructed from binary phase-shift keying (BPSK). Whenonly one of the symbols in SSS and EEE is different, we have d2

E = 4 and when bothsymbols are different from each other, we have d2

E = 8.Fig. 2 displays the ergodic sum-rate of two-way DASTC plotted from (10) for

m0 = m1 = m2 = 2 and γ0 = γ1 = γ2 = 0.5. As can be observed, the analysiscurve match very well with the simulation curve, especially from medium to highSNR regime, which validates the tightness of the proposed approximation. To

124 Part IV

0 5 10 15 20 25 3010-12

10-9

10-6

10-3

100

(m0,m

1,m

2)=(3,3,6)

(m0,m

1,m

2)=(2,4,6)

(m0,m

1,m

2)=(2,3,6)

(m0,m

1,m

2)=(2,2,6)

Analysis Asymptotic

SNR (dB)

Pair

wis

e Er

ror P

roba

bilit

yBPSKd

E

2=4

Figure 4: PWEP of two-way DASTC. Results are shown for γℓ2ℓ=0 = 1/16.

highlight the advantage of two-way over one-way communications, we also plotthe spectral efficiency of one-way DSTC at T2 (detailed proof can be similarlyobtained as in (10)). We see that the two-way system significantly outperformsone-way DASTC, specifically, an increase by 1 nat/s/Hz is observed at 30 dB.

Fig. 3 shows the PWEP performance of three representative scenarios withthe following severity parameters: Scenario 1 mℓ2

ℓ=0 = 3, Scenario 2 mℓ2ℓ=0 =

2, and Scenario 3 mℓ2ℓ=0 = 1. The analytical curves are plotted from (13) and

(15) and results are shown for γℓ2ℓ=0 = 1/16 and d2

E = 4. We can see that thePWEP performance improves as parameter m increases, as expected. In addi-tion, the analysis and simulation results are in good agreement, which verifiesour analysis.

For better understanding the effect of the severity parameter m on the diver-sity performance obtained in (21), Fig. 4 plots the PWEP performance versusaverage SNR for γℓ2

ℓ=0 = 1/16. Here, we select different fading parame-ters in four scenarios, e.g., (m0,m1,m2) = (2, 2, 6), (m0,m1,m2) = (2, 3, 6),(m0,m1,m2) = (2, 4, 6), and (m0,m1,m2) = (3, 3, 6). The diversity order in-creases with parameter m1 while fixing m0 = 2 and m2 = 6 as the slopes ofcurves for the first three scenarios, i.e., (2,2,6), (2,3,6), and (2,4,6), improve.


However, in comparison between the third and the four curves, i.e., (2,4,6) and(3,3,6), they are parallel in the high SNR regime since they have the same valueof m0 + min(m1,m2) = 6, which indicates the obtained diversity order derivedin (21).

5 Conclusion

In this paper, we have derived closed-form expressions for the ergodic sum-rateand PWEP of DASTC transmission in two-way fixed gain AF relay networksover Nakagami-m fading channels. Moreover, to reveal additional insight intothe system performance, the asymptotic expression for PWEP has also been ob-tained. It has been proved that the diversity order solely depends on the fadingseverity parameters as m0 + min(m1,m2). The considered two-way system hasbeen shown to outperform the one-way scheme in terms of spectral efficiencywhile achieving the full diversity gain.


First, we evaluate I1 as

I1 =

∫ ∞

0

∫ ∞

0

ln(αγγ1γ2)fγ1(γ1) fγ2

(γ2) dγ1dγ2

= ln

(

γαγ1γ2

m1m2

)

+ ψ(m1) + ψ(m2) (22)

Then, we calculate I2 as

I2 =

∫ ∞

0

∫ ∞

0

∫ ∞

0

ln

(

1 +γ0

αγ1γ2

)

× fγ0(γ0) fγ1

(γ1) fγ2(γ2) dγ0dγ1dγ2 (23)

We next express ln(1 + γ0

αγ1γ2) in terms of the Meijer’s G-function with the help

of [10, eq. (8.4.6.5)] as follows:

ln

(

1 +γ0

αγ1γ2

)

= G1,22,2

(

γ0

αγ1γ2

∣

∣

∣

∣

∣

1, 1

1, 0

)

(24)

126 Part IV

By substituting (24) to (23) and performing the integral over the variable γ0,we obtain

I2 =

∫ ∞

0

∫ ∞

0

1

Γ(m0)G1,3

3,2

(

γ0

αm0γ1γ2

∣

∣

∣

∣

1 −m0, 1, 11, 0

)

× fγ1(γ1) fγ2

(γ2) dγ1dγ2 (25)

where (25) follows from the help of [9, eq. (7.813.1)]. Then by applying [9,

eq. (9.31.2)], i.e., Gm,np,q

(

x−1

∣

∣

∣

∣

ar

bs

)

= Gn,mq,p

(

x

∣

∣

∣

∣

1 − bs1 − ar

)

, into (25), we get

I2 =

∫ ∞

0

∫ ∞

0

1

Γ(m0)G3,1

2,3

(

αm0

γ0γ1γ2

∣

∣

∣

∣

0, 1m0, 0, 0

)

× fγ1(γ1) fγ2

(γ2) dγ1dγ2 (26)

=

∫ ∞

0

1

Γ(m0)Γ(m1)G3,2

3,3

(

αm0γ1

γ0m1γ2

∣

∣

∣

∣

1 −m1, 0, 1m0, 0, 0

)

× fγ2(γ2) dγ2 (27)

=

G3,34,3

(

αm0γ1γ2

γ0m1m2

∣

∣

∣

∣

1 −m2, 1 −m1, 0, 1m0, 0, 0

)

Γ(m0)Γ(m1)Γ(m2)(28)

where (26), (27), and (28) are followed from the help of [9, eq. (7.813.1)]. Dueto the similarity between J1 and J2, we express Jn with n ∈ 1, 2 as

Jn =

∫ ∞

0

ln (1 + αγn) fγn(γn) dγn

=1

Γ(mn)G1,3

3,2

(

αγn

mn

∣

∣

∣

∣

1 −mn, 1, 11, 0

)

(29)

where (29) is obtained by using [10, eq. (8.4.6.5)] and [9, eq. (7.813.1)]. Byplugging (22), (28), and (29) into (9), we get (10), which finalizes the proof.

Appendix B: Proof of Theorem 2

We have

Mη1s = E

e−sη1

=

∫ ∞

0

∫ ∞

0

∫ ∞

0

e−sαγ1γ2+γ0

αγ2+1

× fγ0(γ0) fγ1

(γ1) fγ2(γ2) dγ0dγ1dγ2. (30)


We first evaluate the integral over γ0 and γ1. Substituting the PDF of γ0 andγ1 from (5) into (30), and using [9, eq (3.381.4)], we obtain

Mη1s =

∫ ∞

0

a0a1

(

s

αγ2 + 1+m0

γ0

)−m0

×(

sαγ2

αγ2 + 1+m1

γ1

)−m1

fγ2(γ2) dγ2 (31)

where ai =(

mi

γi

)mi

, i = 0, 1. After some manipulations, (31) can be rewrittenas

Mη1s =

a0a1mm2

2

Γ(m2)γm2

2

m0+m1∑

k=0

∫ ∞

0

αm0+m1−k (c0γ2 + b0)−m0

× (c1γ2 + b1)−m1 γm0+m1+m2−k−1

2 exp

(

−m2γ2

γ2

)

dγ2. (32)

where c0 = m0αγ0

, c1 = sα+ m1αγ1

, b0 = s+ m0

γ0

, and b1 = m1

γ1

. Define

J ,

∫ ∞

0

xη (d1x+ 1)−ν1 (d2x+ 1)

−ν2 e−µxdx (33)

To evaluate integral J , we first express (d1x+ 1)−ν1 and (d2x+ 1)

−ν2 in termsof Meijer’s G-function by using [10, eq (8.3.2.21)], and then change to Fox H-function with the help of the identity [10, Eq. (8.4.2.5)] as follows:

(1 + dix)−νi =

1

Γ(νi)G1,1

1,1

(

dix

∣

∣

∣

∣

(1 − νi)(0)

)

=1

Γ(νi)H1,1

1,1

[

dix

∣

∣

∣

∣

(1 − νi, 1)(0, 1)

]

(34)

where Hm,np,q [·] is Fox H-function [10, Eq. (8.3.1.1)]. By substituting (34) into

(33) and using [11, Eq. (2.6.2)], we obtain

J =µ−η−1

Γ(ν1)Γ(ν2)H1,1,1,1,1

1,(1:1),0,(1:1)

d1

µ

d2

µ

∣

∣

∣

∣

∣

∣

∣

∣

(1 + η, 1)(1 − ν1, 1) ; (1 − ν2, 1)

—(0, 1) ; (0, 1)

(35)

128 Part IV

where HK,N,N ′,M,M ′

E,(A:C),F,(B:D) [·] is the generalized Fox H-function [11, Eq. (2.2.1)]. By

applying (35) to (32), we obtain

Mη1s =

m0+m1∑

k=0

a0a1αm0+m1−kγm0+m1−k

2

bm0

0 bm1

1 Γ(m0)Γ(m1)Γ(m2)mm0+m1−k2

×H1,1,1,1,11,(1:1),0,(1:1)

c0γ2

b0m2

c1γ2

b1m2

∣

∣

∣

∣

∣

∣

∣

∣

(m0 +m1 +m2 − k, 1)(1 −m0, 1) ; (1 −m1, 1)

—(0, 1) ; (0, 1)

. (36)

Finally, by using [11, Eq. (2.3.2)], (36) can be simplified in the term of Kampede Feriet’s function, and we arrive at the the desired result as in Theorem 2.

References

[1] T. Cui, F. Gao, T. Ho, and A. Nallanathan, “Distributed space-time codingfor two-way wireless relay networks,” IEEE Trans. Signal Process., vol. 57,no. 2, pp. 658–671, Feb. 2009.

[2] Y. Han, S. H. Ting, C. K. Ho, and W. H. Chin, “High rate two-way amplify-and-forward half-duplex relaying with OSTBC,” in Proc. IEEE VTC, Sin-gapore, Apr. 2008, pp. 2426–2430.

[3] ——, “Performance bounds for two-way amplify-and-forward relaying,”IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 432–439, Jan. 2009.

[4] D. To, J. Choi, and I.-M. Kim, “Error probability analysis of bidirectionalrelay systems using Alamouti scheme,” IEEE Commun. Lett., vol. 14, no. 8,pp. 758–760, Aug. 2008.

[5] M. Eslamifar, W. H. Chin, C. Yuen, and G. Y. Liang, “Performance analysisof two-way multiple-antenna relaying with network coding,” in Proc. IEEE

VTC, Anchorage, AK, Sep. 2009, pp. 1–5.

[6] M. Eslamifar, C. Yuen, W. H. Chin, and Y. L. Guan, “Max-min antennaselection for bi-directional multi-antenna relaying,” in Proc. IEEE VTC,Taipei, Taiwan, May 2010.

[7] T. Q. Duong, C. Yuen, H.-J. Zepernick, and X. Lei, “Average sum-rate ofdistributed Alamouti space–time scheme in two-way amplify-and-forward


relay networks,” in Proc. IEEE GLOBECOM Workshop, Miami, FL, Dec.2010, pp. 79–83.

[8] T. Q. Duong, D.-B. Ha, H.-A. Tran, and N.-S. Vo, “Symbol error proba-bility of distributed-Alamouti scheme in wireless relay networks,” in Proc.

IEEE VTC, Marina Bay, Singapore, May 2008, pp. 648–652.

[9] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,6th ed. San Diego, CA: Academic Press, 2000.

[10] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series.New York: Gordon and Breach Science, 1990, vol. 3.

[11] A. M. Mathai and R. K. Saxena, The H-function with Applications in

Statistics and Other Disciplines. New York: Wiley, 1978.

Part V

Part V

Beamforming in Two-Way Fixed Gain


Part V is published as

T. Q. Duong, H. A. Suraweera, H.-J. Zepernick, and C. Yuen, “Beamform-ing in Two-Way Fixed Gain Amplify-and-Forward Relay Systems with CCI,” inProc. IEEE International Communications Conference, Ottawa, Canada, June2012.

Beamforming in Two-Way Fixed Gain


Trung Q. Duong, Himal A. Suraweera, Hans-Jurgen Zepernick,

and Chau Yuen

Abstract

We analyze the outage performance of a two-way fixed gain amplify-and-forward (AF) relay system with beamforming, arbitrary antenna cor-relation, and co-channel interference (CCI). Assuming CCI at the relay,we derive the exact individual user outage probability in closed-form. Ad-ditionally, while neglecting CCI, we also investigate the system outageprobability of the considered network, which is declared if any of the twousers is in transmission outage. Our results indicate that in this system,the position of the relay plays an important role in determining the useras well as the system outage probability via such parameters as signal-to-noise imbalance, antenna configuration, spatial correlation, and CCIpower. To render further insights into the effect of antenna correlationand CCI on the diversity and array gains, an asymptotic expression whichtightly converges to exact results is also derived.

1 Introduction

The commonly assumed one-way relaying protocol in the literature is limitedin system throughput since two time slots are required per single transmission.The loss of system throughput can be recovered by exploiting the concept oftwo-way relay transmission [1]. Consequently, two-way relaying has attracted alot of interest, e.g., [2, 3, 4].

In addition, multiple antenna deployment in fixed gain amplify-and-forward(AF) relay systems can bring further gains at a low practical implementationcomplexity. In particular, hop-by-hop beamforming is a good technique to re-alize the benefits of deploying multiple antennas [5]. However, due to spacelimitation in transmitters and receivers, antenna correlation can degrade the

135

136 Part V

system performance. For one-way beamforming transmission, by consideringantenna correlation at the transceiver, the performance of a dual-hop fixed gainAF relay network over Rayleigh fading channels has been investigated in [5, 6].

Besides antenna correlation, co-channel interference (CCI) presenting in cel-lular systems is another significant impairment that can degrade the systemperformance. Therefore, a substantial number of research works [7, 8, 9, 10]have been carried out to investigate the effect of CCI on the performance offixed gain AF relaying. The outage probability of a fixed gain relay system withCCI at the destination has been reported in [7]. In [8], the effect of multipleRayleigh interferers at the relay has been quantified. In [9] and [10], the com-bined effect of CCI at both the relay and destination on the outage probabilityhas been investigated. However, the systems considered in [9, 10] are limited tosingle antenna model.

In this paper, we consider a two-way relay system in which the source ter-minals are deployed with multiple antennas. Communication between the twousers is facilitated using a single fixed gain AF relay. Furthermore, due to theuse of multiple antennas, we consider beamforming and maximal ratio combin-ing for coherent detection. To the best of the authors’ knowledge, the effect ofCCI on the performance of single or multiple antenna two-way AF relay systemshas not been addressed.

In particular, we derive a closed-form expression for the outage probabilityby considering antenna correlation at user terminals and CCI at the relay. Inorder to gain further insights, we also develop high SNR outage probabilityexpressions. The asymptotic expression reveals that in a fixed and relatively lowinterference environment, antenna correlation of a full rank correlation matrixhas no impact on the diversity gain, which is equal to the minimum number ofantennas equipped at the two sources. In order to comprehensively evaluate theperformance of two-way communication further, we analyze the system outageprobability, where an outage event is declared when any of the two sources isin outage. Thus it is a measure of the overall Quality-of-Service (QoS) that thesystem can offer for two-way communication. We note that the system outagehas not been widely understood even for simple single antenna two-way relaysystems with no CCI effect. The obtained result highlights the significant roleof relay placement on the outage performance.

Notation: The superscripts T and † stand for the transpose and transposeconjugate. ‖yyy‖F denotes Frobenius norm of the vector yyy. EX . is the expecta-tion operator w.r.t. the random variable (RV), X . Γ (n) is the gamma function[11, Eq. (8.310.1)], Γ (a, x) is the incomplete gamma function [11, Eq. (8.350.2)],ψ(x) is Euler psi function [11, Eq. (8.360.1)], Ein (x) =

∫∞

1 e−xt/tndt is the

Beamforming in Two-Way Fixed Gain AF Relay Systems with CCI 137

generalized exponential integral function, and Kn (.) is the nth-order modifiedBessel function of the second kind [11, Eq. (8.432.6)].


Consider a dual-hop two-way AF relay network where two sources S1 and S2

equipped with N1 and N2 antennas, respectively, communicate using a singleantenna AF relay, R. The antenna arrays at S1 and S2 experience spatial an-tenna correlation, which can be characterized by two spatial correlation matricesΞΞΞ1 and ΞΞΞ2, respectively. The communication between S1 and S2 occurs in twohops. In the first hop, S1 and S2 simultaneously transmit two messages s1 ands2 to R. Assume that S1 and S2 have the perfect knowledge for channel stateinformation (CSI) of the links to R, the transmit beamforming can be performedby multiplying s1 and s2 with weighting vectors wwwT1 and wwwT2, respectively. Thesignal received at R, corrupted by L interferes, is given by

yR = hhhT1 ΞΞΞ1/21 wwwT1s1 + hhh2ΞΞΞ

1/22 wwwT2s2 +

L∑

ℓ=1

gℓxℓ + zR, (1)

where hhhn = [hn,1, hn,1, . . . , hn,Nn]T , for n ∈ 1, 2, is the Nn × 1 channel vector

from Sn to R with average channel power Ωn, zR is additive white Gaussian noise(AWGN) with zero mean and variance N0. In (1), xℓ and gℓ are the interferingsignal and channel coefficient from the ℓ-th interferer to R, where E

|xℓ|2

= Pℓand E

|gℓ|2

= Ω3ℓ for ℓ = 1, 2, . . . , L.In the second time-slot, R amplifies yR with the fixed gain G before forwarding

the resulting signal to both S1 and S2. At each source, e.g., S2, the N2 × 1received vector signal yyyS2

is given by

yyyS2= GΞΞΞ

1/22 hhh2ΞΞΞ

1/21 hhh1wwwT1s1 + GΞΞΞ

1/22 hhh2hhh2ΞΞΞ

1/22 wwwT2s2

+ GΞΞΞ1/22 hhh2

L∑

ℓ=1

gℓxℓ + GΞΞΞ1/22 hhh2zR + zzz2, (2)

where zzz2 is the N2 × 1 AWGN vector with zero mean and variance N0. Sinceeach source node knows its own transmitted signal, S2 can remove the self-interference part, i.e., the second term in (2), which requires the knowledge ofγIℓ

= Pℓ/N0 E

|gℓ|2

1, ℓ = 1, . . . , L. We then multiply the received signal with

1R can convey these average values to S1 and S2 and update them periodically using a lowrate feedback channel.

138 Part V

an 1 ×N2 received weighting vector wwwR2 to yield

yS2= wwwR2GΞΞΞ

1/22 hhh2ΞΞΞ

1/21 hhh1wwwT1s1+

+ wwwR2GΞΞΞ1/22 hhh2

L∑

ℓ=1

gℓxℓ + wwwR2GΞΞΞ1/22 hhh2zR + wwwR2zzz2. (3)

According to the principle of hop-by-hop beamforming [8], to maximize thesignal-to-interference plus noise ratio (SINR), transmit weighting vector at S1,i.e., wwwT1, and receive weighting vector at S2, i.e., wwwR2, are chosen as follows:

wwwT1 = (ΞΞΞ12

1 hhh1)/∥

∥

∥ΞΞΞ

12

1 hhh1

∥

∥

∥

F, wwwR2 = (ΞΞΞ

12

2 hhh2)/∥

∥

∥ΞΞΞ

12

2 hhh2

∥

∥

∥

F. (4)

From (3) and (4), the SINR at S2 is expressed as

γS2=

E

|s1|2

∥

∥

∥ΞΞΞ

1/21 hhh1

∥

∥

∥

2

F

∥

∥

∥ΞΞΞ

1/22 hhh2

∥

∥

∥

2

F∥

∥

∥ΞΞΞ

1/22 hhh2

∥

∥

∥

2

F

(

∑Lℓ=1 Pℓ|gl|2 +N0

)

+N0/G2

. (5)

Without loss of generality, we assume that the two sources transmit the sameamount of power, i.e., E

|s1|2

= E

|s2|2

= Ps. To maintain the averagetransmit power at the relay, the amplifying gain G is determined as

1

G2= E

∥

∥

∥ΞΞΞ

12

1 hhh1

∥

∥

∥

2

F+∥

∥

∥ΞΞΞ

12

2 hhh2

∥

∥

∥

2

F+

L∑

ℓ=1

Pℓ

Ps|gℓ|2 + N0

Ps

. (6)

To simplify the notation, we denote γn = γ∥

∥

∥ΞΞΞ1/2n hhhn

∥

∥

∥

2

F, for n ∈ 1, 2, and

γ3 =∑L

ℓ=1 Pℓ/N0|gℓ|2, with γ = Ps/N0. The instantaneous SINR at Sn, forn ∈ 1, 2, can be rewritten as

γSn=

γ1γ2

γn(γ3 + 1) + C , (7)

where C is a constant given by C = Ps/(N0G2), which will be derived in thesequel.

It is important to obtain the statistical characteristics of RV γn, for n ∈1, 2, given in (7). Without loss of generality, assuming that the correlation


matrix ΞΞΞn has Qn distinct non-zero eigenvalues λn1, λn2, . . . , λnQnwith multi-

plicities αn1, αn2, . . . , αnQn, respectively, the probability density function (PDF)

and cumulative distribution function (CDF) of γn can be formulated as

fγn(γ) =

Qn∑

i=1

αni∑

j=1

ϑnijγj−1

Γ(j)(γχni)je− γ

γχni , (8)

Fγn(γ) = 1 −

Qn∑

i=1

αni∑

j=1

j−1∑

k=0

ϑnijk!

(

γ

γχni

)k

e− γ

γχni , (9)

where χni = λniΩn and the expansion coefficient ϑnij is defined as

ϑnij =(χnij)

αni−j

(αni − j)!

dαni−j

dtαni−j

Qn∏

l=1,l 6=i

(t+ χnij)−αni

∣

∣

∣

∣

∣

∣

t=−χnij

.

When the channels are independent, i.e., ΞΞΞn is identity matrix, we have ϑnij = 1for i = 1, j = 1, 2, . . . , Nn and ϑnij = 0 for i = 1, j = 1, 2, . . . , Nn − 1. Whenthe correlation matrix follows an exponential model, i.e., all the eigenvalues

λn1, λn2, . . . , λnNnare distinct, we have ϑni = χNn−1

ni /Nn∏

l=1,l 6=i

(χni − χnl).

Since γ3 is the sum of L exponentially distributed RVs, its PDF is easily

given by fγ3 (γ) =∑Lℓ=1 βℓ exp

(

γγIℓ

)

, where βℓ = γ−1Iℓ

∏

k=1,k 6=ℓ(1− γIk/γIℓ

)−1.

Using the above correlation model, constant C is obtained as

C = γ

Q1∑

i=1

α1i∑

j=1

ϑ1ijjχ1i + γ

Q2∑

r=1

α2r∑

t=1

ϑ2rttχ2r +

L∑

ℓ=1

βℓγ2Iℓ

+ 1. (10)

3 Outage Probability Analysis

The outage event occurs when the instantaneous SINR falls below a predefinedthreshold. In this work, we consider the outage probability in two cases: i) theuser outage probability at S1 or S2 and ii) system outage probability declaredwhen the minimum SINR between S1 and S2 is below a threshold.

3.1 User Outage Probability

In this particular case, the outage probability is defined as the probability thatS1 or S2 is in outage, i.e., Pout = Pr (γSn

< γth) = FγSn(Pout).

140 Part V

3.1.1 Exact Outage Probability

Consider the outage probability for S2. The exact CDF of γS2is given by (see

Appendix A for proof)

Pout = 1 − 2

Q1∑

i=1

α1i∑

j=1

j−1∑

k=0

ϑ1ij

k!(χ1iγ)k

k∑

l=0

(

k

l

)

γkthCk−l

× e−γth/(γχ1i)

Q2∑

r=1

α2r∑

t=1

ϑ2rt

Γ(t)(χ2r γ)tKl+t−k

(

2

√

γthCγ2χ1iχ2r

)

×(

γthCχ2r

χ1i

)

l+t−k2

l∑

s=0

l!

(l − s)!

L∑

ℓ=1

βℓ

(

γth

γχ1i+

1

γIℓ

)−s−1

. (11)

When a particular channel correlation model is adopted, we can further simplify(12). For e.g., in the case of exponential correlation, (11) reduces to

Pout = 1 − 2

N1∑

i=1

ϑ1ie−

γthγχ1i

N2∑

r=1

ϑ2r

χ2rγ

√

γthCχ2r

χ1i

×K1

(

2

√

γthCγ2χ1iχ2r

)

L∑

ℓ=1

βℓ

(

γth

γχ1i+

1

γIℓ

)−1

. (12)

For the case of independent fading, (11) simplifies to

Pout = 1 − 2

N1−1∑

i=0

e−

γthγΩ1

i!(γΩ1)i

i∑

l=0

(il)C

i−lγith

Γ(N2)(γΩ2)N2

(

γthCΩ2

Ω1

)N2+l−i2

×KN2+l−i

(

2√

γthCγ2Ω1Ω2

) l∑

s=0

l!(l−s)!

L∑

ℓ=1

βℓ

(

γth

γΩ1+ 1

γIℓ

)−1

. (13)

3.1.2 Outage Probability at High SNR

To provide additional insights into the behavior of the outage probability andto investigate the diversity order and the array gain of the system, we nowpresent an asymptotic result. In the high signal-to-noise ratio (SNR) regime,i.e., γ → ∞, we can express C ≈ γ where is a constant, which results in (see


Appendix B for proof)

P∞out =

c(N1)

(

γth

γ

)N1

if N1 < N2

c(N3)

(

γth

γ

)N3

ifN1 = N2 = N3

c(N2)

(

γth

γ

)N2

if N1 > N2

, (14)

where constant c(θ), for θ = min(N1, N2), is written as

c(θ) =

Q1∑

i=1

α1i∑

j=1

j−1∑

k=0

ϑ1ij

k!

k∑

l=0

(

k

l

) Q2∑

r=1

α2r∑

t=1

ϑ2rt

Γ(t)

l∑

s=0

l!

(l − s)!

×L∑

ℓ=1

βℓ

(

γ

γχ1i+

1

γIℓ

)−s−1(1

χ1i

)θ

Φ(l, t, k). (15)

Recall that γ = Ps/N0 and γIℓ= Pℓ/N0. In (14), we have assumed that

the interference-to-noise ratio (INR), γIℓ, is low and fixed (γIℓ

does not varywhen the SNR is increased). On the other hand, when a symmetric networkis assumed, such that the interfering terminals transmit with the same powercharacteristics as the useful terminals, (implying that the INR tends to infinitywhen the SNR tends to infinity) e.g., [8, 12], the diversity order becomes zeroregardless of the use of multiple antennas at S1 and S2. Result Φ(l, t, k) in (15)is defined according to the relationship of the running indices l, t, and k asfollows:

• If l + t− k > 0, we have

Φ(l, t, k) =

min(l+t−k−1,θ−k)∑

w=0

(−1)θ−k+1(l + t− k − w − 1)!

w!(θ − k − w)!

×(

χ2r

)k−l+w

+

θ−l−t∑

w=0

(−1)θ−k−w/(θ − t− l − w)!

w!(l + t− k + w)!

×(

χ2r

)t+w [

log

(

γth

γχ1iχ2r

)

− ψ(w + 1)

− ψ(l + t− k + w + 1)

]

. (16)

142 Part V

• If l+ t− k = 0, we have Φ(l, t, k) = Φ(t, k), shown as

Φ(t, k) =

θ−k∑

w=0

log(

γth

γχ1iχ2r

)

− 2ψ(w + 1)

(−1)k+w−θw!w!(θ − k − w)!χt+w2r

. (17)

• If l+ t− k < 0, we have

Φ(l, t, k) =

min(k−l−t−1,θ−t−l)∑

w=0

(−1)θ−l−t+1(k − l − t+ w − 1)!

w!(θ − t− l − w)!

×(

χ2r

)w+t

+

θ−k∑

w=0

(−1)θ−t−l−w

w!(k − l − t+ w)!(θ − k − w)!

×(

χ2r

)k+w−l [

log

(

γth

γχ1iχ2r

)

− ψ(w + 1)

− ψ(k − l− t+ w + 1)

]

. (18)

For an exponential correlation model, c(θ) given in (15) simplifies to

c(θ) =

N1∑

i=1

N2∑

r=1

ϑ1iϑ2r

L∑

ℓ=1

βℓ

χθ1i

(

γ

γχ1i+

1

γIℓ

)−1

Φ1, (19)

where

Φ1 =

θ−1∑

w=0

(−1)θ[

ln(

γthCγχ1iχ2r

)

− ψ(w + 1) − ψ(w + 2)]

w!(w + 1)!(θ − w − 1)!

×(

χ2r

)w+1

+(−1)θ+1

θ!. (20)

From (14), it can be observed that when the channel correlation matrices areof full-rank, the diversity order is equal to the minimum of antennas at S1 andS2, i.e., min(N1, N2), which is the maximum achievable diversity gain, and thatcorrelation does not affect the diversity order.

3.2 System Outage Probability Analysis

The two-way relaying concept considers information exchange between S1 andS2. Therefore, in some applications, successful transmission is declared only


0 5 10 15 20 25 30

10-6

10-5

10-4

10-3

10-2

10-1

100

th=5 dB

N1 =3, N

2=2

1=

2= 0.1, 0.5, 0.9

Closed-form Approximation Simulation

Relay is located half-way between S1 and S

2

SNR (dB)

Out

age

Prob

abili

ty o

f S2

Figure 1: Outage probability of S2 with different correlation coefficients. Resultsare shown for L = 1 and γI1 = 1 dB.

when both S1 and S2 in operation. In other words, the considered system issuspended if any of S1 and S2 is in outage [4]. We define the system outageprobability as

Pout = Pr (min(γS1, γS2

) < γth) . (21)

Since the two RVs γS1and γS2

are dependent, in this case the mathematicalderivation is much involved. Therefore, for the mathematical tractability, weomit the effect of CCI. The instantaneous SINRs given in (7) are now rewrittenas γS1

= γ1γ2γ1+C for S1 and γS2

= γ1γ2γ2+C for S2, which leads to

Pout = Pr [min(γS1, γS2

) < γth]

= Pr (γS1< γth, γS1

< γS2)

︸︷︷︸

I1

+ Pr (γS2< γth, γS2

< γS1)

︸︷︷︸

I2

. (22)

Because of the symmetry between I1 and I2 in (22), we concentrate on I1.

144 Part V

Since the condition γS1< γS2

is equivalent to γ1 > γ2, we obtain

I1 =

∫ ∞

ǫ

fγ1 (γ1)

∫ γth+γthC

γ1

0

fγ2 (γ2) dγ2dγ1

︸︷︷︸

J1

+

∫ ǫ

0

fγ1 (γ1)

∫ γ1

0

fγ2 (γ2) dγ2dγ1

︸︷︷︸

J2

, (23)

where ǫ = 12

(

γth +√

γ2th

+ 4γthC)

is the positive root of the quadratic equation

γ21 − γthγ1 − γthC = 0. The exact expression for I1 is given by (see Appendix C

for detailed proof)

I1 = 1 −Q1∑

i=1

α1i∑

j=1

ϑ1ij

Γ(j)(χ1iγ)j

Q2∑

r=1

α2r∑

t=1

t−1∑

k=0

ϑ2rt

k!

(

γth

γχ2r

)k

×

e−γth/(γχ2r)k∑

l=0

(

k

l

)

Ck−l∞∑

s=0

(−γthCγχ2r

)sǫj+l−s−k

s!

× Eis+k−j−l+1

(

ǫ

γχ1i

)

+

(

1

γχ1i+

1

γχ2r

)−j−k

×[

Γ(j + k) − Γ

(

j + k,ǫ

γχ1i+

ǫ

γχ2r

)]

. (24)

Similarly, I2 is given by

I2 = 1 −Q2∑

i=1

α2i∑

j=1

ϑ2ij

Γ(j)(χ2iγ)j

Q1∑

r=1

α1r∑

t=1

t−1∑

k=0

ϑ1rt

k!

(

γth

γχ1r

)k

×

e−γth/(γχ1r)k∑

l=0

(

k

l

)

Ck−l∞∑

s=0

(−γthCγχ1r

)sǫj+l−s−k

s!

× Eis+k−j−l+1

(

ǫ

γχ2i

)

+

(

1

γχ2i+

1

γχ1r

)−j−k

×[

Γ(j + k) − Γ

(

j + k,ǫ

γχ2i+

ǫ

γχ1r

)]

. (25)

The system outage probability is expressed in the form of one infinite sumwhich is shown to converge very fast. For example, only five number of terms


0 5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

th=5 dB

1 =

2= 0.3

(N1,N

2) = (3,3)

(N1,N

2) = (3,2)

(N1,N

2) = (2,2)

Closed-form Approximation Simulation

Relay is located half-way between S1 and S

2

SNR (dB)

Out

age

Prob

abili

ty o

f S2

Figure 2: Outage probability of S2 with different antenna configurations. Re-sults are shown for L = 3 and γIℓ

3ℓ=1 = 1, 2, 3 dB.

is required in the summation over index s to achieve the accuracy to the degreeof eight decimals.


In this section, we provide numerical results to validate the above analysis. Herewe apply an exponential-decay model for the path loss. Specifically, assume thatthe distance between S1 and S2 is equal to d, we have the corresponding channelmean power Ω0 ∼ d−µ. Then, Ω1 = κ−µΩ0 and Ω2 = (1 − κ)

−µΩ0, where κ

stands for the fraction of the distance from S1 to R over the distance from S1

to S2. For example, when the relay is located in the middle between S1 andS2, we have κ = 0.5. Moreover, an exponential correlation model is used wherethe correlation coefficient between the i-th and j-th antennas of Sn is given by

ρni,j= ρ

|i−j|n with i, j = 1, 2, . . . , Nn and n = 1, 2. In all examples, we set

γth = 5 dB and Ω0 = 1. Unless otherwise stated, R is located half-way betweenS1 and S2, leading to Ω1 = Ω2 = 16.

146 Part V

0 5 10 15 20 25 3010-2

10-1

100

=0.1 =0.5 =0.9

th=5 dB

N1 = N

2 =2

SNR (dB)

Out

age

Prob

abili

ty o

f S2

Figure 3: Effect of CCI on the outage probability of S2 when γIℓ= γ.

Fig. 1 shows the outage probability of S2 for the case of N1 = 3, N2 = 2 anda single interferer with γI1 = 1 dB. As expected, we see that high correlationadversely degrades the outage probability. However, only the array gain isaffected by correlation while the diversity gain remains the same since the threecurves are parallel as plotted in log-log scale. Moreover, the exact curves plottedfrom (11) are in excellent agreement with Monte Carlo simulations and theasymptotic curves plotted from (14) converge to the exact curves.

Fig. 2 shows the outage probability for a fixed correlation coefficient, i.e.,ρ1 = ρ2 = 0.3, and L = 3 with γIℓ

3ℓ=1 = 1, 2, 3 dB. To clearly highlight

the effect of diversity, the antenna configuration at S1 and S2 is selected as(N1, N2) = (2, 2), (3, 2), (3, 3). We notice that for a fixed N1, increasing N2

yields no additional diversity gain as the two systems (2, 2) and (3, 2) have thesame diversity order.

Fig. 3 illustrates the effect of CCI on the system performance by settingγIℓ

proportional to the average SNR as γIℓ= νγ, where ν is a fixed scalar.

We see that, correlation significantly degrades the outage probability as all thecurves exhibit a floor in the high SNR regime. It is interesting to observe thatcorrelation improves the performance in the low SNR region. For low SNR,


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10-2

10-1

100

(N1,N

2) = (2,2)

(N1,N

2) = (2,4)

1 =

2= 0.3

Outage Probability of S2

Outage Probability of S1

System Outage Probability

Relative distance parameter

Out

age

Prob

abili

ty

Figure 4: User outage probability of S1, S2, and system outage probabilityversus the relative position of relay.

correlation allows potentially focused power, which is beneficial to the systemperformance [13]. This explains the observed effect in Fig. 3.

In Fig. 4, in order to show the effect of the relay position, we have plottedthe user and system outage probability versus the relative distance parameterκ for the symmetric case, e.g., (N1, N2) = (2, 2), and asymmetric case, e.g.,(N1, N2) = (2, 4). When R is located nearby S1, we have κ < 0.5. Notice thatS2 outperforms S1 when R is close to S1 and vice versa, which shows that thefirst hop channel governs the user outage probability. In the considered system,interference at R is amplified before forwarding to a user, and it has a dominanteffect when the first-hop link is strong. The system outage probability coincideswith the worst case of S1 and S2, as expected. The best performance for thesymmetric case is achieved at κ = 0.5, while it is interesting to see that in theasymptotic case, the best performance is obtained for κ = 0.3. This observationshows that when the system is balanced, i.e., symmetric topology, R must beplaced in the middle between S1 and S2. However, for the unbalanced case,e.g., (N1, N2) = (2, 4), R must be nearby S1. The shift of relay location to S1

compensates for the imbalance of the system model. Specifically, in our example,

148 Part V

S1 has less number of antennas, hence, in order to achieve a compromise for S1

and S2, a designer must carefully select the system parameters.

5 Conclusions

We have investigated the performance of a two-way fixed gain AF relay systemwith antenna correlation and CCI. In order to analyze the effects of these im-portant practical impairments on the user and system outage probability, wederived the exact closed-form expressions. Moreover, asymptotic result provid-ing further insights into array gain and diversity order was also obtained. Wesee that multiple antenna deployment is an attractive solution to improve theoutage performance when the relay system is affected by low level of CCI. How-ever, if the CCI effect is more severe, the performance significantly deteriorates.As such, multi-antenna interference cancelation schemes could be additionallyimplemented and considered for future work.

Appendix A

The CDF of γS2is written as

FγS2(γ) =

∫ ∞

0

∫ ∞

0

Fγ1

(

γ(γ3 + 1) +Cγγ2

)

× fγ2 (γ2) fγ3 (γ3) dγ2dγ3. (26)

By substituting (8), (9) into (26) and applying the binomial theorem, we obtain

FγS2(γ) = 1 −

Q1∑

i=1

α1i∑

j=1

j−1∑

k=0

ϑ1ij

k!(χ1iγ)k

k∑

l=0

(

k

l

)

γkCk−l

× e−γ/(γχ1i)

Q2∑

r=1

α2r∑

t=1

ϑ2rt

Γ(t)(χ2rγ)t

L∑

ℓ=1

βℓ

×∫ ∞

0

∫ ∞

0

γl−k+t−12 exp

(

− Cγγχ1iγ2

− γ2

γχ2r

)

× (γ3 + 1)l exp

(

− γγ3

γχ1i− γ3

γIℓ

)

dγ2dγ3. (27)


To solve the above double integral, we first expand (γ3 + 1)l into a finite sumby using binomial theorem [11, Eq. (1.111)], and then use [11, Eq. (4.471.9)].After some simplifications, we obtain (11).

Appendix B

Due to the space limit, we will briefly introduce the approach to calculate the

asymptotic result. First, we expand the exponential term e−γthγχ1i into the infinite

sum using Taylor expansion. Second, we employ series representation for Besselfunction as

Kν (z) =

ν−1∑

w=0

Γ(ν − w)

Γ(w + 1)

(−1)w

2

(z

2

)−ν+2w

+ (−1)ν+1∞∑

w=0

( z2 )ν+2w[

ln( z2 ) − ψ(w+1)2 − ψ(ν+w+1)

2

]

Γ(w + 1)Γ(ν + w + 1). (28)

Depending on the relationship of the running indices l, t, and k, ν = l + t − kcan be positive, negative, or zero. When ν > 0 we utilize (28). When ν < 0, wefirst apply Kν (·) = K−ν (·) and then (28). For zero value, i.e., l+ t− k = 0, thefollowing expansion is valid

K0 (z) = − lnz

2

∞∑

w=0

(

z2

)2w

(w!)2+

∞∑

w=0

z2w

22w(w!)2ψ(w + 1). (29)

Then by substituting the partial coefficients ϑnij , for the terms γnth

, the sumof these terms becomes zero when n < θ. Therefore, the lowest order of theexponent n is equal to θ.

Appendix C

We first consider J1 given in (23) as

J1 =

∫ ∞

ǫ

fγ1 (γ1)Fγ2

(

γth +γthCγ1

)

dγ1. (30)

150 Part V

Now, by substituting (8), (9) into (30), and performing some manipulationsyields

J1 = 1 − Fγ1 (ǫ) −Q1∑

i=1

α1i∑

j=1

ϑ1ij

Γ(j)(χ1i γ)j

Q2∑

r=1

α2r∑

t=1

t−1∑

k=0

ϑ2rt

k! e−

γth

γχ2r

×(

γth

γχ2r

)k∫ ∞

ǫ

γj−k−11 (γ1 + C)ke

−γ1γχ1i e

−γthC

γχ2rγ1 dγ1. (31)

To the best of our knowledge, the integral in (31) has no closed-form solution.To solve this integral, we first apply binomial theorem [11, Eq. (1.111)] for term(γ1 + C)k and using Taylor series representation [11, Eq. (1.211.1)] for term

e−γthC

γχ2rγ1 , which results in

J1 = 1 − Fγ1 (ǫ) −Q1∑

i=1

α1i∑

j=1

ϑ1ij

Γ(j)(χ1iγ)j

Q2∑

r=1

α2r∑

t=1

t−1∑

k=0

ϑ2rt

k!

×(

γth

γχ2r

)k

e−γth

γχ2r

k∑

l=0

(

k

l

)

Ck−l∞∑

s=0

(−γthCγχ2r

)s1

s!

×∫ ∞

ǫ

γj+l−k−s−11 e

−γ1

γχ1i dγ1. (32)

The integral representation in (32) can be obtained in the form of the generalizedexponential integral function Eiν (·).

Next, we evaluate J2 given in (23) by rewriting it as

J2 =

∫ ǫ

0

fγ1 (γ1)Fγ2 (γ1) dγ1

= Fγ2 (ǫ) −Q1∑

i=1

α1i∑

j=1

ϑ1ij

Γ(j)(χ1iγ)j

Q2∑

r=1

α2r∑

t=1

t−1∑

k=0

ϑ2rt

k!

1

(γχ2r)k

×∫ ǫ

0

γj+k−11 e

−γ1

γχ1−

γ1γχ2r dγ1. (33)

The above integral can be solved in the form of gamma function Γ(n) andincomplete gamma function Γ(n, x) with the help of [11, Eq. (3.381.8)]. Finally,we sum (32) and (33) and with rearrangement to arrive at (24).


References

[1] Y. Han, S. H. Ting, C. K. Ho, and W. H. Chin, “Performance bounds fortwo-way amplify-and-forward relaying,” IEEE Trans. Wireless Commun.,vol. 8, no. 1, pp. 423–439, Jan. 2009.

[2] R. H. Y. Louie, Y. Li, and B. Vucetic, “Practical physical layer networkcoding for two-way relay channels: performance analysis and comparison,”IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 764–777, Feb. 2010.

[3] J. Yang, P. Fan, T. Q. Duong, and X. Lei, “Exact performance of two-wayAF relaying in Nakagami-m fading environment,” IEEE Trans. Wireless


[4] P. K. Upadhyay and S. Prakriya, “Performance of two-way opportunis-tic relaying with analog network coding over Nakagami-m fading,” IEEE

Trans. Veh. Technol., vol. 60, no. 4, pp. 1965–1971, May 2011.

[5] H. A. Suraweera, H. K. Garg, and A. Nallanathan, “Beamforming in dual-hop fixed gain relay systems with antenna correlation,” in Proc. IEEE ICC,Cape Town, South Africa, May 2010, pp. 1–5.

[6] N. S. Ferdinand and N. Rajatheva, “Unified performance analysis of two-hop amplify-and-forward relay systems with antenna correlation,” IEEE

Trans. Wireless Commun., 2011, in press.

[7] C. Zhong, S. Jin, and K.-K. Wong, “Dual-hop systems with noisy relay andinterference-limited destination,” IEEE Trans. Commun., vol. 58, no. 3, pp.764–768, Mar. 2010.

[8] H. A. Suraweera, D. S. Michalopoulos, R. Schober, G. K. Karagiannidis,and A. Nallanathan, “Fixed gain amplify-and-forward relaying with co-channel interference,” in Proc. IEEE ICC, Kyoto, Japan, Jun. 2011, pp.1–6.

[9] A. M. Cvetkovic, G. T. D. Ordevic, and M. C. Stefanovic, “Performance ofinterference-limited dual-hop non-regenerative relays over Rayleigh fadingchannels,” IET Commun., vol. 5, no. 2, pp. 135–140, 2011.

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152 Part V

[11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,6th ed. San Diego, CA: Academic Press, 2000.

[12] I. Krikidis, J. S. Thompson, S. McLaughlin, and N. Goertz, “Max-min relayselection for legacy amplify-and-forward systems with interference,” IEEE

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[13] A. M. Tulino, A. Lozano, and S. Verdu, “Impact of antenna correlation onthe capacity of multiantenna channels,” IEEE Trans. Inf. Theory, vol. 51,no. 7, pp. 2491–2509, Jul. 2005.

Part VI-A

Part VI-A

Exact Outage Probability of Cognitive AF

Relaying with Underlay Spectrum Sharing

Part VI-A is published as

T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick, “Exact Outage Probabilityof Cognitive AF Relaying with Underlay Spectrum Sharing,” Electron. Lett.,vol. 47, no. 47, pp. 1001-1002, Aug. 2011.

Exact Outage Probability of Cognitive AF

Relaying with Underlay Spectrum Sharing

T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick

Abstract

The exact closed-form expression for the outage probability (OP) ofcognitive radio dual-hop amplify-and-forward relay networks is derived.The tractable expression of the OP, given in the form of elementary func-tions, readily enables us to evaluate the effect of primary users on thesecondary system performance. It has been shown that the use of AF re-laying significantly improves the performance of cognitive radio networkscompared to its direct transmission counterpart.

1 Introduction

Cognitive radio with spectrum sharing has significantly improved the spectrumefficiency by allowing secondary users (SUs) to simultaneously share the fre-quency band licensed to primary users (PUs) without causing any harmful inter-ference on PUs. The performance analysis for cognitive radio with applicationsto relay networks has gained great attention in the research community [1–8].In particular, the outage probability for cognitive decode-and-forward (DF) re-lay networks has been presented in [1–3]. Recently, the issue of relay selectionfor cognitive DF relay networks have been addressed in [4–7]. It is importantto note that all of the above mentioned works, i.e., [1–7], only considered DFrelays. Very recently, the authors in [8] have taken into account the amplify-and-forward (AF) relays to investigate the performance of cognitive relay networks.However, this analysis utilized the bounded signal-to-noise ratio (SNR), i.e., theend-to-end SNR for AF relays is approximated as the minimum SNR among thetwo hops. As a result, the problem to investigate cognitive AF relay networksequivalently becomes opportunistic DF relays. More importantly, the analysisin [8] has been conducted for the high SNR regime, which is not appropriatefor cognitive networks with underlay spectrum sharing (requiring an acceptable

157

158 Part VI-A

level of interference on PUs). To the best of our knowledge, there has been noprevious works considering the performance of cognitive AF relay networks.

Inspired by all of the above, in this paper, we derive the exact outage prob-ability for cognitive AF relay networks over non-identical Rayleigh fading chan-nels. Our final outage expression is given in a compact form and validated byMonte-Carlo simulations. Utilizing the analytical expression, we can evaluatethe impact of PUs on SUs’ systems and highlight the advantage of using AFrelays for cognitive radio networks over direct transmission.


We consider a dual-hop spectrum-sharing system with the coexistence of PUsand SUs by sharing the same narrow-band frequency as shown in Fig. 1. In thesecondary network, for the first hop transmission, the SU transmitter (SU-Tx)sends signal x to the SU relay (SU-Relay). To ensure that the SU transmissiondoes not cause any harmful interference on PUs, the transmit power at SU-TxPS is set at PS = Ip/|hs,p|

2, where Ip is the maximum tolerable interferencepower at PU and hs,p is the channel coefficient of the link from SU-Tx to PU.As a result, the received signal at the SU-Relay is given by

yr = hs,rx+ nr (1)

where hs,r is the channel coefficient for the link from SU-Tx to SU-Relay and nr isadditive white Gaussian noise (AWGN). Then, the received signal at SU-Relayis amplified with variable gain G and forwarded to the SU receiver (SU-Rx).Due to power constraint, the SU-Relay should limit its transmitted power toPR = Ip/|hr,p|

2, where hr,p is the channel coefficient from SU-Relay to PU. Thereceived signal at SU-Rx is given by

yd = Ghr,dhs,rx+Ghr,dnr + nd (2)

where nd is AWGN at the SU-Rx. In this paper, we consider non-identicalRayleigh fading for all links in which the channel power gain |hu,v|

2 is expo-nentially distributed with E

|hu,v|2

= Ωu,v, where u ∈ s, r, v ∈ r, p, d,and E · denotes the expectation. We further assume that all AWGN compo-nents have zero mean and variance N0. To derive the amplifying gain G, wewill utilize the fact that PR = G2(|hs,r|

2PS + N0). Since PS = Ip/|hs,p|2 and

PR = Ip/|hr,p|2, we can obtain G from the following expression

1/G2 = |hr,p|2

(

|hs,r|2

|hs,p|2

+N0

Ip

)

(3)

Exact OP of Cognitive AF Relaying with Underlay Spectrum Sharing 159

SU-

Tx

PU

SU-

Relay

SU-

Rx

interference link data link

r,dhs,rh

s,phr,ph

Figure 1: System model for cognitive amplify-and-forward relay networks.

The end-to-end SNR at SU-Rx after the maximum likelihood decoding can beexpressed as

γd =G2

|hr,d|2|hs,r|

2PS

G2|hr,d|

2N0 +N0=

Ip

N0

|hs,r|2

|hs,p|2Ip

N0

|hr,d|2

|hr,p|2

Ip

N0

|hs,r|2

|hs,p|2+

Ip

N0

|hr,d|2

|hr,p|2+ 1

. (4)

3 Exact Outage Probability Analysis

It is important to note that the exact SNR expression give in (4) has not ap-peared in the literature. For notational simplification, we write (4) as

γd =γ1γ2

γ1 + γ2 + 1(5)

where γ1 =Ip

N0

|hs,r|2

|hs,p|2and γ2 =

Ip

N0

|hr,d|2

|hr,p|2. To obtain the statistical characteris-

tics of γd, we need to find the probability density function (PDF) fγn(γ) and

cumulative distribution function (CDF) Fγn(γ) for n ∈ 1, 2. In other words,

we need to find the PDF and CDF of Z = αXY , where α is a positive constant;

X and Y are two exponentially distributed random variables with parametersΩx and Ωy, respectively. The CDF of Z can be obtained from

FZ (z) =

∫ ∞

0

FX

(yz

α

)

fY (y) dy = 1 − (1 + λz)−1 (6)

160 Part VI-A

where λ =Ωy

αΩx. Taking the derivative of FZ (z) with respect to z yields

fZ (z) = λ(1 + λz)−2 (7)

With the obtained statistics for γn, we are now able to derive the CDF of γd as

Fγd(γ) = Fγ2 (γ) +

∫ ∞

γ

Pr(

γ1 ≤γγ2+γγ2−γ

)

fγ2 (γ2) dγ2 (8)

By applying the change of variable t = γ2 − γ for the integral, after somealgebraic manipulations, the CDF of γd can be given by

Fγd(γ) = 1 −

∫ ∞

0

λ2tdt

[λ1γ(γ + 1) + (1 + λ1γ)t] (1 + λ2γ + λ2t)2 , (9)

where λ1 = (IpΩs,p)/(Ωs,rN0) and λ2 = (IpΩr,p)/(Ωr,dN0). To proceed furtherthe calculation, we need to solve the following integral

I =

∫ ∞

0

ϕ(x)dx

ψ(x)(10)

where ϕ(x) = β4x, ψ(x) = (β2 + β1x)(β3 + β4x)2, and β1, β2, β3, β4 > 0. To do

so, let us rewrite the integrand of I into the partial expansion as follows:

ϕ(x)

ψ(x)=

1

β1β4

[

A1

(x+ β2/β1)+

B1

(x + β3/β4)+

B2

(x+ β3/β4)2

]

, (11)

where A1, B1, and B2 are partial coefficients given by

A1 =−β2/β1

(

β3

β4−

β2

β1

)2 , B1 =β2/β1

(

β3

β4−

β2

β1

)2 , B2 =β3/β4

(

β3

β4−

β2

β1

) ,

which then yields,

I =1

β1β3 − β2β4+β2β4 ln

(

β2β4

β1β3

)

(β1β3 − β2β4)2. (12)

Finally, from (12) and (9), the CDF of γd can be obtained in closed-form ex-pression as follows:

Fγd(γ) = 1 −

1

(1 + λ1γ)(1 + λ2γ) − λ1λ2γ(γ + 1)

−

λ1λ2γ(γ + 1)

[(1 + λ1γ)(1 + λ2γ) − λ1λ2γ(γ + 1)]2ln

[

λ1λ2γ(γ + 1)

(1 + λ1γ)(1 + λ2γ)

]

.

(13)


The outage probability Pout is defined as the probability that the end-to-endSNR γd falls below a given threshold γth. As a result, we have Pout = Fγd

(γth),which immediately follows from (13).

4 Numerical Results

We consider a linear dual-hop AF relay network in a 2-D plane, where all SUsare located in a straight line. Furthermore, the SU-Tx and SU-Rx are locatedat coordinates (0,0) and (1,0), respectively, and their distance is normalized toone. The SU-Relay node is placed in half-way between SU-Tx and SU-Rx, i.e.ds,r = dr,d = 1/2. The pass loss of each link follows an exponential-decay model.In other words, the average channel power for the transmission between node uand node v is modeled as Ωu,v = 1/dǫ

u,v where ǫ denotes the path loss exponentwith u ∈ s, r and v ∈ p, r, d. For a typical non-line-of-sight propagationmodel, we can set ǫ = 4. To evaluate the effect of PU on SU’s networks, weconsider three different scenarios in which PU is located at different coordinates(0.44, 0.44), (0.55, 0.55), and (0.66, 0.66). We also compare the performance ofcognitive AF relay networks and conventional cognitive radio networks, i.e, onlydirect transmission from SU-Tx to SU-Rx. Hence, it is convenient to providethe outage probability for cognitive radio with direct transmission as (detailedderivations are omitted here due to space limit)

P(DT)out =

γth

γth +IpΩs,d

N0Ωs,p

. (14)

As can be observed from Fig. 2, the analytical results obtained from the exactclosed-form outage probability given in (13) match very well with simulations,which validates our analysis. For comparison, the performance of cognitiveAF relaying substantially outperforms that of direct transmission for all threeexamples. Approximately, by using the AF relay, a gain of 4.5 dB can beobtained as compared to the direct transmission. In addition, the position of PUsignificantly affects the performance of SU’s networks. The best performancecan be achieved for the case when PU is located at coordinate (0.66, 0.66),in which the minimum interference power on PU is satisfied among the threerepresentative scenarios.

162 Part VI-A

-15 -10 -5 0 5 10 15 20

10-2

10-1

100

PU(0.66,0.66) Exact closed-form Simulation Direct transmision

PU(0.55,0.55)

PU(0.44,0.44)

Exact closed-form Simulation Direct transmision

th = 1 dB

Exact closed-form Simulation Direct transmision

Ip/N

0 (dB)

Out

age

Prob

abili

ty

Figure 2: Outage probability of cognitive amplify-and-forward relay networks.

5 Conclusion

We have derived the exact closed-form expression for the outage probability ofcognitive AF relay networks under interference power constraint. Our analyt-ical results are valid for non-identical Rayleigh fading channels and provide apowerful tool to assess the effect of PU on the performance of cognitive radionetworks with relaying assistance. It has been certified that the dual-hop relay-ing is a promising candidate for cognitive radio networks since its performancesurpasses the conventional cognitive radio direct transmission.

References

[1] L. Luo, P. Zhang, G. Zhang, and J. Qin, “Outage performance for cogni-tive relay networks with underlay spectrum sharing,” IEEE Commun. Lett.,vol. 15, no. 7, pp. 710–712, Jul. 2011.

[2] Y. Guo, G. Kang, N. Zhang, W. Zhou, and P. Zhang, “Outage perfor-mance of relay-assisted cognitive-radio system under spectrum-sharing con-


straints,” IET Electron. Lett., vol. 46, no. 2, pp. 182–184, Jan. 2010.

[3] S. Yan, X. Wang, and H. Zhang, “Performance analysis of the cognitivecooperative scheme based on cognitive relays,” in IEEE Inter. Conf. on

Commun. Workshops (ICC), Cape Town, South Africa, May 2010.

[4] J. Lee, H. Wang, J. G. Andrews, and D. Hong, “Outage probability of cog-nitive relay networks with interference constraints,” IEEE Trans. Wireless

Commun., vol. 10, no. 2, pp. 390–395, Feb. 2011.

[5] L. Li, X. Zhou, H. Xu, G. Y. Li, D. Wang, and A. Soong, “Simplified relayselection and power allocation in cooperative cognitive radio systems,” IEEE

Trans. Wireless Commun., vol. 10, no. 1, pp. 33–36, Jan. 2011.

[6] J. Si, Z. Li, X. Chen, B. Hao, and Z. Liu, “On the performance of cognitiverelay networks under primary user’s outage constraint,” IEEE Commun.

Lett., vol. 15, no. 4, pp. 422–424, Apr. 2011.

[7] V. Asghari and S. Aissa, “Cooperative relay communication performanceunder spectrum-sharing resource requirements,” in IEEE Inter. Conf. on

Commun. (ICC), Cape Town, South Africa, May 2010.

[8] H. Ding, J. Ge, D. B. d. Costa, and Z. Jiang, “Asymptotic analysis of cooper-ative diversity systems with relay selection in a spectrum sharing scenario,”IEEE Trans. Veh. Technol., vol. 60, no. 2, pp. 457–472, Feb. 2011.

Part VI-B

Part VI-B

Cooperative Spectrum Sharing Networks with

AF Relay and Selection Diversity

Part VI-B is published as

T. Q. Duong, V. N. Q. Bao, G. C. Alexandropoulos, and H.-J. Zepernick, “Co-operative Spectrum Sharing Networks with AF Relay and Selection Diversity,”Electron. Lett., vol. 47, no. 20, pp. 1149–1151, Sep. 2011.

Cooperative Spectrum Sharing Networks with AF

Relay and Selection Diversity

T. Q. Duong, V. N. Q. Bao, G. C. Alexandropoulos,

and H.-J. Zepernick

Abstract

The exact outage probability (OP) of cognitive dual-hop relay net-works equipped with a single amplify-and-forward (AF) relay and a se-lection combining receiver at the destination is derived under spectrumsharing constraint on primary user. The tractable closed-form OP readilyenables us to evaluate the system performance, which indicates the sig-nificance of using relay in cognitive radio networks with spectrum sharingapproach. The proposed analysis is validated by numerical examples.

1 Introduction

Cognitive radio technology is a promising approach to improve the utilizationof scarce radio frequency spectrum. The concept of relaying communication incognitive radio networks with cooperative spectrum sharing has attracted greatattention [1–3]. All of these works investigated the cognitive relay networkswith decode-and-forward (DF) relaying operation. In particular, for adaptiveDF relays, the exact and tight bounds for outage probability (OP) were derivedin [1] and [2], respectively. Besides DF, amplify-and-forward (AF) relay is alsoan important protocol where the relay just simply forwards the signal to thedestination without performing any regenerating operation. Recently, the OPof cognitive relay networks with AF relay has been derived in [4]. By approxi-mating the end-to-end signal-to-noise ratio (SNR) as the harmonic mean of tworandom variables, the tight lower bound of OP was derived in the high SNRregime. However, it is important to note that such approximation is not feasiblein practical cognitive radio scenario since the underlay spectrum sharing systemmust operate in the low SNR regime to avoid causing any harmful interference

169

170 Part VI-B

on the primary networks. To circumvent this drawback, the closed-form ex-pression of exact OP for cognitive AF relay networks has been presented in [5].However, the work in [5] has not considered the direct link of secondary networkswhich can be utilized for diversity combining with relaying link to increase thenetwork reliability.

Inspired by all of the above, our paper presents for the first time the co-operative spectrum sharing with AF relay and selection combining (SC). Inthe cooperative spectrum sharing model, the existence of a common randomvariable, i.e., the channel fading coefficient from SU transmitter (SU-Tx) to pri-mary user (PU), results in a dependence between the two SNR terms, which iscumbersome for the analysis. This paper proposes a new analytical approachto solve this problem by taking into account the conditioned statistics on thefading coefficient from SU-Tx to PU. Specifically, the exact OP is obtained ina tractable closed-form expression (containing only elementary functions). Thisresult readily allows us to investigate the advantage of deploying AF relay in acognitive spectrum sharing environment. In fact, the performance of the consid-ered cognitive cooperation significantly outperforms both direct communicationsand AF relaying transmission without direct link.


In this paper, we consider a dual-hop spectrum-sharing system with the co-existence of PUs and SUs as shown in Fig. 1. The secondary relay network,consisting of an SU-Tx, an SU-Relay, and a SU receiver (SU-Rx), can operatein the same spectrum licensed to the PU as long as the SU transmission doesnot cause any harmful interference on the PU. For the first hop transmission,SU-Tx broadcasts signal s to both SU-Relay and SU-Rx with the maximumtransmitted power PS given as PS = Ip/|hs,p|2, where Ip is the maximum toler-able interference power at PU and hs,p is the channel coefficient of the link fromSU-Tx to PU. Then, the received signal at SU-Relay is amplified with variablegain G and forwarded to SU-Rx. Due to the power constraint, SU-Relay shouldlimit its transmitted power to PR = Ip/|hr,p|2, where hr,p is the channel coef-ficient from SU-Relay to PU. In this paper, we consider non-identical Rayleighfading for all links in which the channel power gain |hA,B|2 is exponentiallydistributed with variance ΩA,B, where A ∈ s, r, B ∈ r, p, d, and AWGNcomponents have zero mean and variance N0.

Cooperative Spectrum Sharing Networks with AF Relay and Selection Diversity 171

SU-Tx

PU

SU-

Relay

SU-Rx

r,dhs,rh

s,phr,ph

interference link data link

s,dh

Selection

Combining

Figure 1: System model for cognitive amplify-and-forward relay networks.

3 Exact Outage Probability Analysis

The instantaneous received SNR after SC is defined as

γsc = max(γaf , γdt) (1)

where γaf and γdt are SNRs of relaying and direct links, respectively, and aregiven by [4, 5]

γaf =

IpN0

|hs,r|2|hs,p|2

IpN0

|hr,d|2|hr,p|2

IpN0

|hs,r|2|hs,p|2 +

IpN0

|hr,d|2|hr,p|2 + 1

, γdt =Ip

N0

|hs,d|2|hs,p|2

. (2)

where hs,r and hs,d are the channel coefficients for the link from SU-Tx to SU-Relay and to SU-Rx. As can be observed from (2), the two terms γaf andγdt have a common value |hs,p|2, leading to the statistical dependence amongthem. As a result, the cumulative distribution function (CDF) of γsc can notbe similarly obtained as in conventional relay systems, i.e.,

Fγsc (γ) 6= Fγaf(γ)Fγdt

(γ) (3)

due to the appearance of |hs,p|2 in both γaf and γdt. To tackle this problem,thanks to the statistical independence between hs,p and other terms in (2), i.e.,hs,r, hr,d, hr,p, and hs,d, the CDF of γsc conditioned on hs,p can be written as

Fγsc (γ|hs,p) = Fγaf(γ|hs,p)Fγdt

(γ|hs,p) (4)

172 Part VI-B

We now aim to derive the conditional statistics of γaf given in (2). By denoting

y =IpN0

|hs,r|2, γ1 =IpN0

|hr,d|2|hr,p|2 , and x = |hs,p|2, we have

Fγaf(γ|hs,p) = Pr

( yxγ1

yx + γ1 + 1

< γ

∣

∣

∣

∣

hs,p

)

= 1 − λ1e−λs,rγx

∫ ∞

0

e−λs,r(γ2+γ)x/t

(1 + λ1γ + λ1t)2dt. (5)

where λ1 =Ωr,p

IpΩr,d/N0and λs,r = N0/(IpΩs,r). By applying the change of variable

z = 1/t and with the help of [6, eq. (3.353.3)], [6, eq. (8.359.1)], (5) can becomputed as

Fγaf(γ|hs,p) = 1 − e−λs,rγx

1 + λ1γ+

λ1λs,r(γ2 + γ)x

(1 + λ1γ)2

× exp[(

λ1λs,r(γ2+γ)

1+λ1γ − λs,rγ)

x]

Γ(

0,λ1λs,r(γ

2+γ)x1+λ1γ

)

, (6)

where Γ(·, ·) is the incomplete gamma function [6, eq. (8.350.2)]. In addition,the CDF of γdt conditioned on hs,p as shown in (2) can be easily derived as

Fγdt(γ|hs,p) = 1 − e−λs,dγx (7)

where λs,d = N0/(IpΩs,d). The unconditional CDF of γsc marginalized outw.r.t. hs,p is therefore given by

Fγsc (γ) =

∫ ∞

0

Fγaf(γ|hs,p)Fγdt

(γ|hs,p) f|hs,p|2 (x) dx (8)

By plugging everything together and taking the expectation over |hs,p|2, theCDF of γsc can be expressed as Fγsc (γ) = 1 + J1 + J2 − J3, where

J1 =1

(1 + λ1γ)[Ωs,p(λs,r + λs,d)γ + 1]− 1

1 + Ωs,pλs,dγ

− 1

(1 + λ1γ)(1 + λs,rΩs,pγ). (9)

and J2, J3 are two integrals, respectively, given by

J2 =

∫ ∞

0

θxe−(λs,rγ+1/Ωs,p)xeθxΓ(0, θx)

(1 + λ1γ)Ωs,pdx,

J3 =

∫ ∞

0

θxe−(λs,rγ+λs,dγ+1/Ωs,p)xeθxΓ(0, θx)

(1 + λ1γ)Ωs,pdx, (10)


where θ =λ1λs,r(γ

2+γ)1+λ1γ . The two above integrals, i.e., J2 and J3 , have a similar

form as

J =

∫ ∞

0

αxe−βxeαxΓ(0, αx)dx (11)

To compute (11), we utilize [6, eq. (6.455.1)], resulting in

J =α

2β2 2F1

(

1, 2; 3;β − α

β

)

, (12)

where 2F1 (a, b; c; z) is the Gauss hypergeometric function [6, eq. (9.111)]. Next,using Gauss’s recursion function [6, eq. (9.137.4)], we can express

2F1 (1, 2; 3; z) =2

z[2F1 (1, 1; 2; z)− 2F1 (0, 2; 2; z)] (13)

From the facts that 2F1 (0, 2; 2; z) = 1 (see, [6, eq. (9.121.1)]) and 2F1 (1, 1; 2; z) =

− log(1−z)z (see, [6, eq. (9.121.6)]), we obtain a more simplified expression for J

as

J =α

β(α − β)+

α log(β/α)

(α − β)2. (14)

Pulling everything together, we can obtain the closed-form expression for Fγsc (γ)as

Fγsc (γ) = 1 − 1

1 + Ωs,pλs,dγ− 1

(1 + λ1γ)(1 + λs,rΩs,pγ)

+1

(1 + λ1γ)[Ωs,p(λs,r + λs,d)γ + 1]

+λ1λs,r(γ

2 + γ)

λ1λs,r(γ2 + γ) − (1 + λ1γ)(λs,rγ + Ω−1s,p)

×

1λs,rγ+Ω−1

s,p+

(1 + λ1γ) log(

(1+λ1γ)(λs,rγ+Ω−1s,p)

λ1λs,r(γ2+γ)

)

λ1λs,r(γ2 + γ) − (1 + λ1γ)(λs,rγ + Ω−1s,p)

− λ1λs,r(γ2 + γ)

λ1λs,r(γ2 + γ) − (1 + λ1γ)(λs,rγ + λs,dγ + Ω−1s,p)

×

1λs,rγ+λs,dγ+Ω−1

s,p+

(1 + λ1γ) log(

(1+λ1γ)(λs,rγ+λs,dγ+Ω−1s,p)

λ1λs,r(γ2+γ)

)

λ1λs,r(γ2 + γ) − (1 + λ1γ)(λs,rγ + λs,dγ + Ω−1s,p)

(15)

174 Part VI-B

As a result, the OP Pout, i.e., the probability that the instantaneous SNR isbelow a certain threshold γth, can be immediately obtained as Pout = Fγsc (γth).

4 Numerical Results

We consider the co-linear topology for all nodes. Assume that the channel meanpower for the link from node A to node B is characterized as ΩA,B = l−

A,B withA ∈ s, r and B ∈ p, r, d, where l is the distance and denotes the pathloss exponent and can be set to four for a typical non-line-of-sight propagationmodel. SU-Relay is located half-way in between SU-Tx and SU-Rx, leading tols,r = lr,d = ls,d/2. By setting the coordinates of SU-Tx and SU-Rx as (0,0) and(1,0), respectively, we obtain ls,d = 1 and ls,r = 1/2. Here, we assume γth = 1dB and PU’s location as (0.55,0.55).

-15 -10 -5 0 5 10 15 20

10-3

10-2

10-1

100

PU(0.55,0.55)

th = 1 dB

Cooperative spectrum sharing (Analysis) Cooperative spectrum sharing (Simulation) Direct transmission Relaying link only

Ip/N

0 (dB)

Out

age

Prob

abili

ty

Figure 2: Performance comparison for different transmission strategies.

Fig. 2 compares the OP performance for different transmission scenarios ofcognitive spectrum sharing including the cooperative spectrum sharing with SC,the relaying spectrum sharing (no direct transmission), and the direct spectrumsharing (only direct transmission). The use of a relay in cognitive networks is


significantly beneficial as the performance of the AF relaying is improved by 5 dBcompared to the direct transmission at the OP below 10−1. However, only thecoding gain is enhanced since the diversity order for the two cases (AF withoutSC and direct transmission) is one. By exploiting the cooperative diversity, wecan remarkably enhance the OP performance as the diversity gain is increasedto the factor of two compared to both relaying link and direct transmission.Finally, an excellent match between analytical results and simulations validatesthe correctness of our analysis.

5 Conclusion

The performance of cooperative spectrum sharing with AF and SC has beenconsidered in this paper. We have derived an exact closed-form expression forOP over non-identical Rayleigh fading channels. The exact OP is expressedin terms of elementary functions, which allows us to evaluate the system per-formance and highlights the advantage of deploying cooperative diversity insecondary networks compared to relaying-only and direct transmission.

References

[1] Y. Guo, G. Kang, N. Zhang, W. Zhou, and P. Zhang, “Outage perfor-mance of relay-assisted cognitive-radio system under spectrum-sharing con-straints,” IET Electron. Lett., vol. 46, no. 2, pp. 182–184, Jan. 2010.



Lett., vol. 15, no. 4, pp. 422–424, Apr. 2011.

[4] H. Kim, H. Wang, J. Lee, S. Park, and D. Hong, “Outage probability of cog-nitive amplify-and-forward relay networks under interference constraints,” inProc. APCC, Auckland, New Zealand, Nov. 2010, pp. 373–376.

[5] T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick, “Exact outage probabilityof cognitive AF relaying with underlay spectrum sharing,” IET Electron.

Lett., vol. 47, no. 17, pp. 1001–1002, Aug. 2011.

176 Part VI-B


Part VI-C

Part VI-C

Effect of Primary Networks on the Performance

of Spectrum Sharing AF Relaying

Part VI-C is published as

T. Q. Duong, V. N. Q. Bao, H. Tran, G. C. Alexandropoulos, and H.-J. Zeper-nick, “Effect of Primary Networks on the Performance of Spectrum Sharing AFRelaying,” Electron. Lett., vol. 48, no. 1, pp. 25–27, Jan. 2012.

Effect of Primary Networks on the Performance

of Spectrum Sharing AF Relaying

T. Q. Duong, V. N. Q. Bao, H. Tran, G. C. Alexandropoulos,

and H.-J. Zepernick

Abstract

Most of the research in spectrum sharing has neglected the effect ofinterference from primary users. In this paper, we investigate the per-formance of spectrum sharing amplify-and-forward relay networks underinterference-limited environment, where the interference induced by thetransmission of primary networks is taken into account. In particular, aclosed-form expression tight lower bound of outage probability is derived.To reveal additional insights into the effect of primary networks on thediversity and array gains, an asymptotic expression is also obtained.

1 Introduction

Spectrum sharing relay networks have recently attracted great attention for pro-viding higher reliability over direct transmission under scarce and limited spec-trum conditions [1–4]. Specifically, the performance of decode-and-forward (DF)relay networks in spectrum sharing environments has been reported in [1–3]. Re-cently, we have investigated the outage probability (OP) for spectrum sharingnetworks with amplify-and-forward (AF) relaying in [4]. It has been shownin [1–4] that utilizing DF/AF relaying significantly enhances the system perfor-mance in such constrained transmission power conditions. However, most of theprevious works have neglected the effect of the primary transmitter (PU-Tx),which significantly deteriorates the performance of the secondary network. Inthis paper, to evaluate this interference effect, we derive a closed-form expres-sion for OP and further calculate an asymptotic expression. We have shownthat under fixed interference from primary networks, the diversity order re-mains unchanged and the loss only occurs in the array gain, which is theoreti-cally quantified. However, when the interference is linearly proportional to the

181

182 Part VI-C

signal-to-noise ratio (SNR) of the secondary network, the system is severelyaffected, i.e., leading to an irreducible error floor of OP.

2 System Model and Outage Probability Ana-

lysis

Consider an underlay cognitive network where a secondary transmitter (SU-Tx) communicates with a secondary receiver (SU-Rx) through the assistanceof a secondary relay (SU-Relay) in co-existence with a primary network asshown in Fig. 1. The transmit powers at the SU-Tx and the SU-Relay areconstrained so that their transmission will not cause any harmful interferenceto the PU-Rx, which is defined by the maximum tolerable interference powerIp. In the first hop, the SU-Tx transmits its signal, s, to the SU-Relay under

the power constraint that PS =Ip

|g1|2 , where g1 is the channel coefficient for the

link SU-Tx→PU-Rx. The received signal at the SU-Relay, yr, impaired by thetransmission of the PU-Tx, is given by

yr =√

PSh1s +√

PIf1x1 + nr (1)

where h1 is the channel coefficient for the link SU-Tx→SU-Relay, PI is theaverage transmit power at the PU-Tx, x1 is the transmitted signal of the PU-Tx in the first time slot, and nr is additive white Gaussian noise (AWGN) at theSU-Relay. Without loss of generality, we assume that E

|s|2

= E

|x1|2

=1, where E · is the expectation. Then, the SU-Relay amplifies yr with anamplifying gain G and transmits the resulting signal to the SU-Rx with theaverage power PR =

Ip|g2|2 , where g2 is the channel coefficient for the link SU-

Relay→PU-Rx. Due to the concurrent transmission of the PU-Tx, the receivedsignal at the SU-Rx can be written as

yd =√

PSGh2h1s + Gh2nr + Gh2

√

PIf1x1 + nd +√

PIf2x2 (2)

where h2 and f1 are the channel coefficients for the links SU-Relay→SU-Rxand PU-Tx→SU-Rx, respectively, x2 is the transmitted signal of the PU-Txwith E

|x2|2

= 1, and nd is AWGN at the SU-Rx. In this work, we con-sider non identical Rayleigh fading in which all the fading channel coefficientsh1, h2, g1, g2, f1, f2 are complex Gaussian distributed with zero mean and vari-ances Ωh1 , Ωh2 , Ωg1 , Ωg2 , Ωf1 , Ωf2 , respectively, and AWGN components nr, nd

have the same variance of N0. The signal-to-interference ratio at the SU-Tx is

Effect of Primary Networks on the Performance of Spectrum Sharing AF Relaying 183

SU Tx

SU

Relay

SU Rx

PU Tx PU Rx

1h

2h

1g

2g

1f

2f

SU SU:

SU PU:

PU SU:

Figure 1: System model for the spectrum sharing AF relay network consideringthe interference from the PU-Tx.

obtained as

γAF =

γ|h1|2|g1|2γI |f1|2

γ|h2|2|g2|2γI |f2|2

γ|h1|2|g1|2γI |f1|2 + γ|h2|2

|g2|2γI |f2|2 + 1(3)

where γ =IpN0

and (3) is obtained by considering the interference-limited envi-

ronment, i.e., γI = PI

N0.

3 Outage Probability Analysis

To start our analysis, let us introduce an upper bound for γAF given in (3) as

γAF ≤ γAFup= min (γ1, γ2) (4)

where

γ1 =γ|h1|

2

|g1|2γI |f1|

2and γ2 =

γ|h2|2

|g2|2γI |f2|

2(5)

In order to obtain the OP, we need to derive the CDF of U = XY Z

where X, Yand Z are exponentially distributed random variables with parameters λx, λy,

184 Part VI-C

and λz , respectively. It is easy to see that the CDF of U can be obtained as

FU (u) =

∫ ∞

0

∫ ∞

0

FX (uyz) fY (y) fZ (z)dydz (6)

Here, the CDF and probability density function (PDF) of W ∈ X, Y, Z arewritten as FW (w) = 1 − e−λww and fW (w) = λwe−λww for λw ∈ λx, λy, λz.After some simple calculations, the CDF of U can be easily derived as

FU (u) = 1 −

λyλz

λxuexp

(

λyλz

λxu

)

Γ

(

0,λyλz

λxu

)

(7)

where Γ(·, ·) is the incomplete gamma function [5, eq. (8.350.2)]. As a result, theCDF of γAFup

, i.e., FγAFup(γ) = 1 − [1 − Fγ1 (γ)] [1 − Fγ2 (γ)], can be written

as

FγAFup(γ) = 1 −

γ2Ωh1Ωh2

γ2IΩg1Ωf1

Ωg2Ωf2γ2 e

(

γΩh1

γIΩg1Ωf1γ

)

e

(

γΩh2

γIΩg2Ωf2γ

)

× Γ(

0,γΩh1

γIΩg1Ωf1γ

)

Γ(

0,γΩh2

γIΩg2Ωf2γ

)

(8)

The lower bound for OP, Pout, can be immediately obtained from (8) utilizing thefact that Pout = FγAFup

(γth), where γth is an outage threshold. The asymptoticrepresentation of Γ(a, x) for large value of |x| can be given by [5, eq. (8.357.1)]

Γ(0, x) = x−1e−x

[

M−1∑

m=0

(−1)mm!

xm+ O

(

|x|−M)

]

, M = 1, 2, . . . ,∞ (9)

By substituting this result into (8) and neglecting small terms, we obtain

Pout

γ→∞≈

(

Ωg1Ωf1

Ωh1

+Ωg2Ωf2

Ωh2

)

γIγth

γ(10)

For comparison, we also derive an asymptotic expression for the case of neglect-ing the effect of the PU-Tx in [4], i.e., in the absence of γI , Ωf1 , and Ωf2 . Thelower bound for OP is shown as (detailed proof is omitted here due to spacelimitation)

Pout = 1 −

(

1 +Ωg1

Ωh1 γγth

)−1 (

1 +Ωg2

Ωh2 γγth

)−1

(11)


0 5 10 15 20 25 30

10-2

10-1

100

PU-Tx(0.9,0.9)

PU-Tx(0.8,0.8)

PU-Tx(0.7,0.7)

PU-Rx(0.5,0.5)

th = 3 dB

Analysis Simulation Asymptotic

SNR, (dB)

O

utag

e Pr

obab

ility

Figure 2: Performance comparison for different positions of the PU-Tx.

Then, applying the McLaurin series expansion for (1 + ax)−1

=∑∞

k=0(−1)kakxk,after some manipulations and ignoring small terms, the asymptotic OP of thesystem in [4] is shown as

Pout

γ→∞≈

(

Ωg1

Ωh1

+Ωg2

Ωh2

)

γth

γ(12)

From (10), i.e., in the presence of the PU-Tx, and (12), i.e., in the absenceof the PU-Tx, we observe that under a fixed γI , the two systems have thesame diversity order. However, the array gain is reduced by an amount of

G∞ = 10 log10

(

(Ωg1Ωf1Ωh2

+Ωg2Ωf2Ωh1)γI

Ωg1Ωh2+Ωg2Ωh1

)

. When the inference from the PU-

Tx, γI , is linearly proportional to the average SNR, i.e., γI = ργ where ρ is apositive constant, the OP in (8) becomes

Pout

γ→∞,γI=ργ≈ ρ

(

Ωg1Ωf1

Ωh1

+Ωg2Ωf2

Ωh2

)

γth (13)

which is independent of γ. This causes an error floor in the OP for the wholeSNR range yielding zero diversity order.

186 Part VI-C

0 5 10 15 20 25 30

10-2

10-1

100

PU-Tx(0.6,0.6)

PU-Rx(0.5,0.5)

I=2 dB

I=4 dB

I=6 dB

I =0.5

I =0.1

No PU-Tx signal

th = 3 dB

SNR, (dB)

Out

age

Prob

abili

ty

Figure 3: Performance comparison for different average powers from the PU-TxγI .

4 Numerical Results

Similarly as in [4], a linear network topology is assumed here where the SU-Tx, the SU-Relay, and the SU-Rx are located at co-ordinates (0, 0) (0, 1

2 ),and (1, 0), respectively. The average channel power for the link between nodeA and B, Ω0, is inversely proportional to the distance from A to B, d0, i.e,Ω0 = 1

d40

for a shadowed urban cellular radio, where A, B ∈ SU-Tx, SU-Relay,

SU-Rx, PU-Tx, PU-Rx. The outage threshold γth is set to 3 dB for all exam-ples. Fig. 2 displays the OP performance for PU-Rx(0.5, 0.5) and γI = 2 dB.Here, we consider three different scenarios where the location of the PU-Tx is setto (0.7, 0.7), (0.8, 0.8), and (0.9, 0.9). As expected, the performance increaseswhen the PU-Tx moves away from the secondary network, i.e, (0.7, 0.7) →

(0.8, 0.8) → (0.9, 0.9). The analysis matches very well with the simulation andthe asymptotic result tightly converges to the exact value, which validates theproposed analysis. To better understand the impact of the PU-Tx on the systemperformance, Fig. 3 shows OP for different values of the interference power γI .In the case of γI being independent of the average SNR γ, i.e., γI = 2, 4, 6 dB,


increasing γI degrades the array gain but not the diversity gain. The PU-Tx hasa major impact on the secondary network since the performance loss of morethan 10 dB is observed in case of the interference of γI = 2 dB compared tothe scenario without the PU-Tx. More severely, as γI = 0.1γ and γI = 0.5γ,the performance is significantly reduced due to the error floor for the consideredSNR range.

5 Conclusion

The effect of the primary network on spectrum sharing AF relaying has beeninvestigated in this paper. Closed-form and asymptotic expressions for OP havebeen derived for non-identical Rayleigh fading channels. It has been shown thatunder a fixed interference from the primary network, the diversity order of thesecondary network is not affected but only the array gain. However, when theinterference power is dependent on the average SNR of the secondary network, itis infeasible to operate the secondary system as an irreducible error floor existsfor the whole SNR regime.

References

[1] D. da Costa, H. Ding, and J. Ge, “Interference-limited relaying transmissionsin dual-hop cooperative networks over Nakagami-m fading,” IEEE Commun.

Lett., vol. PP, no. 99, pp. 1–3, 2011.


Lett., vol. 15, no. 4, pp. 422–424, Apr. 2011.


[4] T. Q. Duong, V. N. Q. Bao, and H.-J. Zepernick, “Exact outage probabilityof cognitive AF relaying with underlay spectrum sharing,” IET Electron.

Lett., vol. 47, no. 17, pp. 1001–1002, Aug. 2011.


Blekinge Institute of TechnologyDoctoral Dissertation Series No. 2012:08

School of Computing

On the PerfOrmance analysis Of cOOPerative cOmmunicatiOns with Practical cOnstraints

On

th

e P

er

fO

rm

an

ce

an

aly

sis

Of

c

OO

Pe

ra

tiv

e c

Om

mu

nic

at

iOn

s w

ith

Pr

ac

tic

al

cO

ns

tr

ain

ts

Quang Trung D

uong

ISSN 1653-2090

ISBN 978-91-7295-235-5

With the rapid development of multimedia services,

wireless communication engineers may face a ma-

jor challenge to meet the demand of higher data-rate

communication over error-prone mobile radio chan-

nels. As a promising solution, the concept of coope-

rative communication, where a so-called relay node

is formed to assist the direct link, has recently been

applied to alleviate the severe pathloss and shado-

wing effects in wireless systems. In addition, wit-

hout spending extra spectrum and power resources,

multiple-input multiple-output (MIMO) antenna

systems have been shown to provide an immense

improvement in system performance compared to

its single-antenna counterpart. As such, cooperative

MIMO communication is essential for wireless and

mobile networks because of its remarkable increase

in spectral efficiency and reliability. Although the

utilization of cooperative communication in MIMO

systems has gained great attention in the literature,

most of the research works have assumed perfect

conditions. Inspired by the aforementioned discus-

sion, this thesis takes a step further to investigate the

performance of cooperative communications with

practical constraints. The thesis provides a general

framework for performance analysis of cooperative

communications subject to several practical cons-

traints such as antenna correlation, rank-deficiency

of the channel matrix, co-channel interference, and

interference-limited constraint of cognitive radio

networks based on an underlay spectrum-sharing

approach.

The thesis is divided into six parts. The first part

investigates the performance of orthogonal space-

time block codes (OSTBCs) over MIMO relay

networks in Nakagami-m fading channels under

the antenna correlation effect. The second part ex-

tends the full-rank MIMO channel to the case of the

MIMO channel matrix being of rank-deficiency. Se-

veral important findings on the impact of the sing-

le-keyhole effect (SKE) and double-keyhole effect

(DKE) are observed for two types of amplifying

mechanism at the relay, namely, linear and squaring

approaches. An important observation corroborated

by our studies is that for offering a tradeoff bet-

ween performance and complexity, we should use

the linear approach for SKE and the squaring ap-

proach for DKE. The third part generalizes the key-

hole effect to multi-keyhole channels. The exact and

asymptotic expressions for symbol error probability

(SEP) are derived for some specific cases such as

multi-keyhole MIMO/multiple-input single-output

(MISO) channel. The fourth part proposes a distri-

buted Alamouti space-time code for two-way fixed

gain amplify-and-forward (AF) relaying. In parti-

cular, closed-form expressions for approximated er-

godic sum-rate and exact pairwise error probability

(PWEP) are derived for Nakagami-m fading chan-

nels. To reveal further insights into array and diver-

sity gains, an asymptotic PWEP is also obtained.

The fifth part analyzes the outage performance of a

two-way fixed gain AF relay system with beamfor-

ming, arbitrary antenna correlation, and co-channel

interference (CCI). Finally, the sixth part investiga-

tes the impact of interference power constraint on

the performance of cognitive relay networks based

on the spectrum-sharing approach.

aBstract

2012:08

2012:08

Quang Trung Duong

on the performance analysis of o cooperative...

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